Next Talk
Complete Schedule
Friday, February 3
Gabor Kun, Courant Institute
A measurable version of the Lovasz Local Lemma.
Abstract:
The Lovasz Local Lemma (LLL) is one of the
basic tools in probabilistic combinatorics. The LLL
was only proved for discrete probability spaces. We
will prove a measurable version of the LLL. To see
what this means consider the following easy corollary
of the LLL:
Given S_1, ..., S_m r-element subsets of the real
numbers, where m<2^r/2e(r+1) the real numbers
have a 2-coloring s.t. there is no monochromatic
translate of any S_i.
The original LLL does not guarantee nice color
classes: the Axiom of Choice is used. We will see
how to do this in a measurable way. We apply this
measurable lemma to the dynamical von Neumann
problem highlighting an interesting connection
to percolation theory.
Warren Weaver Hall Room 512 at 9:50 am.
Friday, February 10
Tom LaGatta, Courant Institute
Geodesics of Random Riemannian Metrics.
Abstract:
Geodesics are local length-minimizing paths in Riemannian
geometry, but it is an interesting question under what conditions they
globally minimize length. The Cartan-Hadamard theorem, for example,
says that under non-positive curvature assumptions on one's space,
geodesics are globally minimizing. In the context of a random metric,
one expects a presence of positive curvature, and random geodesics
should occasionally run into these positive patches. For perturbations
of the Euclidean plane, we have used the point-of-view of the particle
technique to show that this is indeed the case, and that a geodesic
with randomly selected starting conditions is not minimizing (almost
surely). This is joint work with Janek Wehr.
Warren Weaver Hall Room 512 at 9:50 am.
Friday, February 17
Alexey Shashkin, Moscow State University
Limit theorems for geometrical characteristics of Gaussian excursion sets.
Abstract:
Excursion sets of stationary random fields have attracted much
attention in recent years. They have been applied to modeling
complex geometrical structures in tomography, astrophysics and
hydrodynamics. Given a random field and a specified level, it is
natural to study geometrical functionals of excursion sets
considered in some bounded observation window. Main examples of such
functionals are the volume, the surface area and the Euler
characteristics. Starting from the classical Rice formula (1945),
many results concerning calculation of moments of these geometrical
functionals have been proven. There are much less results concerning
the asymptotic behavior (as the window size grows to infinity), as
random variables considered here depend non-smoothly on the
realizations of the random field. In the talk we discuss several
recent achievements in this domain, concentrating on asymptotic
normality and functional central limit theorems.
Warren Weaver Hall Room 512 at 9:50 am.
Friday, February 24
H. T. Yau, Harvard University
TBA.
Abstract:
TBA
Warren Weaver Hall Room 512 at 9:50 am.
Friday, March 2
Columbia-Princeton Probability Day.
Confirmed Speakers:
J. C. Mattingly (Duke University)
R. Pemantle (University of Pennsylvania)
L. Saloff-Coste (Cornell University)
T. Seppäläinen (University of Wisconsin-Madison)
M. Damron (Princeton University)
Please visit: http://www.math.columbia.edu/~fjv/PS/CPPD12/ to register and for more information.
Friday, March 9
TBA, TBA
TBA.
Abstract:
TBA
Warren Weaver Hall Room 512 at 9:50 am.
Friday, March 16
No Seminar. Spring Recess.
Friday, March 23
TBA, TBA
TBA.
Abstract:
TBA
Warren Weaver Hall Room 512 at 9:50 am.
Friday, March 30
TBA, TBA
TBA.
Abstract:
TBA
Warren Weaver Hall Room 512 at 9:50 am.
Friday, April 6
Toby Johnson, TBA
TBA.
Abstract:
TBA
Warren Weaver Hall Room 512 at 9:50 am.
Friday, April 13
TBA, TBA
TBA.
Abstract:
TBA
Warren Weaver Hall Room 512 at 9:50 am.
Friday, April 20
TBA, TBA
TBA.
Abstract:
TBA
Warren Weaver Hall Room 512 at 9:50 am.
Friday, April 27
TBA, TBA
TBA.
Abstract:
TBA
Warren Weaver Hall Room 512 at 9:50 am.
Friday, May 4
TBA, TBA
TBA.
Abstract:
TBA
Warren Weaver Hall Room 512 at 9:50 am.
Fall Semester 2011
Minerva Foundation Lectures at Columbia University, September 7--15
Denis Talay, INRIA Sophia Antipolis
Model Risk: Modeling, Analysis, Control and Numerics.
Abstract:
The objective of these lessons is to show that model risk analysis, particularly financial model risk analysis, opens new interesting stochastic analysis problems, to present recent mathematical and numerical techniques to tackle them, and to analyze mathematically some robust strategies which, issued from the technical analysis, do not rely on a specific mathematical model. We will also present a selection of challenging open questions.
Various theories will be used, such as statistics of random processes, stochastic control, Malliavin calculus, backward stochastic differential equations, viscosity solutions of nonlinear Partial Differential equations. However the course will be self-contained and, whenever possible, the proofs will be fully detailed.
More information at the announcement.
Friday, September 9
Denis Talay, INRIA Sophia Antipolis
Stochastic Approaches for Parabolic and Elliptic Diffraction Equations.
Abstract:
We consider partial differential equations of parabolic or elliptic type
involving a divergence form operator with a discontinuous coefficient and a compatibility transmission condition.
We prove existence and uniqueness results by stochastic methods
which also allow us to develop or justify low complexity Monte Carlo numerical resolution methods
and to get sharp convergence rate estimates.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, September 16 - Two consecutive talks.
Helmut Katzgraber, Texas A&M University and ETH Zurich
Universality in Levy spin glasses.
Abstract:
Spin glasses are paradigmatic models that deliver concepts relevant
for a variety of systems. Concepts from the solution of the mean-field
model, such as ergodicity breaking, aging and ultrametricity have been
applied to realistic short-range spin-glass models as well as to fields
as diverse as structural biology, geology, computer science and even
financial analysis. However, despite ongoing research spanning several
decades in the area of glassy systems, there remain many fundamental
open questions. Rigorous analytical results are difficult to obtain
for spin-glass models, in particular for realistic short-range
systems. Therefore large-scale numerical simulations are the tool of
choice. After presenting a brief overview of spin glasses, the concept
of universality, a cornerstone of statistical physics, is discussed.
Although it is well established numerically that universality is
not violated for nearest-neighbor spin glasses with compact disorder
distributions (e.g., Gaussian and bimodal), some studies suggest that
this might not be the case when the disorder distributions are broad,
as in the case of the Levy distribution. Using large-scale Monte
Carlo simulations that combine parallel tempering with specialized
cluster moves, as well as innovative scaling techniques, we show that
Levy spin glasses do obey universality.
Work done in collaboration with J. C. Andresen and K. Janzen.
Warren Weaver Hall Room 317 at 10:00 am.
Amir Dembo, Stanford University
Lecture series on "Gibbs measures and phase transitions on sparse random graphs"
Abstract:
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We will review this approach and provide some results towards a rigorous treatment of these problems.
Outline for Day 1 of 4:
Warren Weaver Hall Room 317 at 11:00 am.
Friday, September 23 - Two consecutive talks.
Dmitry Ioffe, Technion
Critical drifts for random walks in attractive potentials.
Abstract:
Self-attractive random walks (polymers) undergo a phase transition in
terms of the applied drift: If the drift is strong enough, then the walk
is ballistic, whereas in the case of small drifts self-attraction wins and
the walk is sub-ballistic. We show that, in any dimension larger than
one,
this transition is of first order. In fact, we prove that the walk is
already ballistic at critical drifts, and establish the corresponding LLN
and CLT.
Joint work with Yvan Velenik.
Warren Weaver Hall Room 317 at 10:00 am.
Amir Dembo, Stanford University
Lecture series on "Gibbs measures and phase transitions on sparse random graphs"
Abstract:
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We will review this approach and provide some results towards a rigorous treatment of these problems.
Outline for Day 2 of 4:
Warren Weaver Hall Room 317 at 11:00 am.
Friday, September 30 - Two consecutive talks.
Mykhaylo Shkolnikov, Stanford University
On diffusions interacting through their ranks.
Abstract:
We will discuss systems of diffusion processes on the real line, in which the dynamics of every single process is determined by its rank in the entire particle system. Such systems arise in mathematical finance and statistical physics, and are related to heavy-traffic approximations of queueing networks. Motivated by the applications, we address questions about invariant distributions, convergence to equilibrium and concentration of measure for certain statistics, as well as hydrodynamic limits and large deviations for these particle systems. Parts of the talk are joint work with Amir Dembo, Tomoyuki Ichiba, Soumik Pal and Ofer Zeitouni.
Warren Weaver Hall Room 317 at 10:00 am.
Amir Dembo, Stanford University
Lecture series on "Gibbs measures and phase transitions on sparse random graphs"
Abstract:
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We will review this approach and provide some results towards a rigorous treatment of these problems.
Outline for Day 3 of 4:
Warren Weaver Hall Room 317 at 11:00 am.
Friday, October 7 - Two consecutive talks.
Olivier Bernardi, MIT
Computing the moments of the GOE bijectively.
Abstract:
The GOE, or Gaussian Orthogonal Ensemble, is a Gaussian measure on the set of orthogonal matrices. We consider the problem of finding the nth moment of the eigenvalues of the matrices in the GOE. It turns out that this problem is closely related to a question about the different ways of gluing the edges of a 2n-gon in pairs so as to create a surface without boundary. More precisely, among the (2n)!/n! possible gluings, how many times does one get each surface (considered up to homeomorphism)?
In this talk, we will recall the connection between the two questions, and present a bijective solution. Our results are analogous to the one obtained by Harer and Zagier (1986) about the gluings of a 2n-gon giving an orientable surface (or in matrix terms, about the Gaussian Unitary Ensemble). We also recover a recurence formula for the moments of the GOE recently obtained by Ledoux.
Warren Weaver Hall Room 317 at 10:00 am.
Amir Dembo, Stanford University
Lecture series on "Gibbs measures and phase transitions on sparse random graphs"
Abstract:
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We will review this approach and provide some results towards a rigorous treatment of these problems.
Outline for Day 4 of 4:
Warren Weaver Hall Room 317 at 11:00 am.
Friday, October 14 - Two consecutive talks.
Vladas Sidoravicius, IMPA
From random interlacements to coordinate and infinite cylinder
percolation.
Abstract:
During the talk I will focus on the connectivity properties of
three models with long (infinite)
range dependencies: Random Interlacements, percolation of the vacant set
in infinite
rod model and Coordinate percolation. The latter model have polynomial
decay in sub-critical and super-critical regime in dimension 3.
I will explain the nature of this phenomenon and why it is difficult to
handle these models technically. In the second half
of the talk I will present key ideas of the multi-scale analysis which
allows to reach some conclusions. At the end I will discuss
applications and several open problems.
Warren Weaver Hall Room 317 at 10:00 am.
Ohad Feldheim, Tel Aviv University
Rigidity of 3-colorings of the d-dimensional discrete torus.
Abstract:
We prove that a uniformly chosen proper coloring of Z_{2n}^d
with 3 colors has a very rigid structure when the dimension d is
sufficiently high. The coloring takes one color on almost all of either
the even or the odd sub-lattice. In particular, one color appears on
nearly half of the lattice sites. This model is the zero temperature case
of the 3-states anti-ferromagnetic Potts model, which has been studied
extensively in statistical mechanics. The proof involves results about
graph homomorphisms and various combinatorial methods, and follows a
topological intuition.
Joint work with Ron Peled.
Warren Weaver Hall Room 317 at 11:05 am.
Friday, October 21 - Two consecutive talks.
Amir Dembo, Stanford University
Factor models on locally tree-like graphs
Abstract:
Consider factor (graphical) models on sparse graph sequences
that converge locally to a random tree T. Using a novel interpolation
scheme we prove existence of limiting free energy density under
uniqueness of relevant Gibbs measures for the factor model on T.
We demonstrate this for Potts and independent sets models and further
characterize this limit via large-deviations type minimization
problem and provide an explicit formula for its solution, as the Bethe
free energy for a suitable fixed point of the belief propagation recursions
on T (thereby rigorously generalize heuristic calculations by statistical
physicists using ``replica'' or ``cavity'' methods).
This talk is based on a joint work with Andrea Montanari and Nike Sun.
Warren Weaver Hall Room 317 at 10:00 am.
Ivan Corwin, Microsoft Research - New England, MA
Brownian Gibbs line ensembles.
Abstract:
The Airy line ensemble arises in scaling limits of growth models, directed polymers, random matrix theory, tiling problems and non-intersecting line ensembles. This talk will mainly focus on the "non-intersecting Brownian Gibbs property" for this infinite ensemble of lines. Roughly speaking, the measure on lines is invariant under resampling a given curve on an interval according to a Brownian Bridge conditioned to not intersect the above of below labeled curves. This property leads to the proof of a number of previously conjectured results about the top line of this ensemble. We will also briefly touch on the KPZ line ensemble, which arises as the scaling limit of a diffusion defined by the Doob-h transform of the quantum Toda lattice Hamiltonian. The top labeled curve of this KPZ ensemble is the fixed time solution to the famous Kardar-Parisi-Zhang stochastic PDE. This line ensemble has a "softer" Brownian Gibbs property in which resampled Brownian Bridges may cross the lines above and below, but at exponential energetic cost.
Warren Weaver Hall Room 317 at 11:05 am.
Friday, October 28 - Two consecutive talks.
Giambattista Giacomin, Université Paris Diderot
Coherence stability and effect of random natural frequencies in populations of coupled oscillators.
Abstract:
We consider the (noisy) Kuramoto model, that is a population of N oscillators, or rotators, with mean-field interaction. Each oscillator has its own natural frequency,
which is chosen randomly (quenched disorder) and it is stirred by Brownian motion. In the limit of large N this model is accurately described by a (deterministic) Fokker-Panck equation. We study this equation and obtain quantitatively sharp results in the limit of weak disorder. We show that, in general, the oscillators synchronize (for sufficiently strong interaction) around a common rotating phase, whose frequency is sharply estimated. These results are obtained by identifying the stable hyperbolic manifold of stationary solutions of an associated non disordered model and by exploiting the robustness of hyperbolic structures under suitable perturbations. The method applies also to cases in which the single rotator dynamics is not just a (random) rotation: in fact, to a certain extent, the single rotator dynamics can be arbitrary.
Warren Weaver Hall Room 317 at 10:00 am.
Edward Waymire, Oregon State University
Tree polymers under weak/strong disorder.
Abstract:
Tree polymers are simplifications of
1+1 dimensional lattice polymers made up
of polygonal paths of a (nonrecombining)
binary tree having random path probabilities.
As in the case of lattice polymers, the path
probabilities are (normalized) products of i.i.d.
positive weights. The a.s. probability laws of these
paths are of interest under weak and strong types of
disorder. Some recent results, speculation and conjectures
will be presented for this class of models under both weak
and strong disorder conditions. In particular results are included
that suggest an explicit formula for the asymptotic variance of
the ``free end'' under strong disorder. This is based on joint
work with Stanley Williams and Torrey Johnson.
Warren Weaver Hall Room 317 at 11:05 am.
Tuesday, November 1 - Special talk
Lorenzo Bertini, Universita' di Roma La Sapienza
Large deviation principle of the empirical current for Markov processes.
Abstract:
We consider a continuous time Markov chain on a countable state space and extend the classical Donsker-Varadhan large deviation principle for the empirical measure by considering also the empirical flow. We then discuss the application to the Gallavotti-Cohen functional, whose associated large deviation principle can be obtained by projection. We finally illustrate briefly the analogous results for diffusion processes on R^n.
Warren Weaver Hall Room 1314 at 2:00 pm.
Friday, November 4 - Two consecutive talks.
Roberto Cyril, Université de Marne-La-Vallée
Some rigorous result on the East Model.
Abstract:
We will consider a special example of one dimensional kinetically constrained model, the East model.
We will start by briefly reviewing some of the known results on the dynamics : spectral gap, persistence function, long-time behavior starting from non-equilibrium.
Then, we will focus on the low temperature non-equilibrium dynamics which follows a quench from an initial distribution which is different from the reversible one. This setting has been extensively studied in physics literature: on the basis of heuristic arguments and numerical simulations it was observed that dynamics can be approximated by an irreversible coarsening process for the domains (intervals separating consecutive vacancies) with a peculiar hierarchical structure. We will explain how, provided the initial distribution of the domains is a renewal process, this approximation can be made rigorous and how, by analyzing the asymptotic behavior of the coalescence process, one can prove a staircase behavior for the persistence function, an aging behavior for the correlation function and give a sharp description on the statistics of the intervals separating consecutive vacancies.
(based on a series of papers in collaboration with N. Cancrini, A. Faggionato, F. Martinelli and C. Toninelli).
Warren Weaver Hall Room 317 at 10:00 am.
Nike Sun, Stanford University
Potts and independent set models on d-regular graphs.
Abstract:
We consider the ferromagnetic Potts on typical d-regular graphs, and the independent set model on typical bipartite d-regular graphs, with graph size tending to infinity. We show that the replica symmetric (Bethe) prediction applies for all parameter values in these two models. In this talk I will describe some of the proof techniques, which will give an indication of the contrast with the anti-ferromagnetic Potts model and the independent set model at high fugacity on non-bipartite graphs, where the Bethe prediction is known to fail.This is joint work with Amir Dembo, Andrea Montanari, and Allan Sly.
Warren Weaver Hall Room 317 at 11:05 am.
Wednesday, November 9 - Special talk
Alan Sokal, New York University
Some wonderful conjectures at the boundary between analysis, combinatorics and probability.
Abstract:
I discuss some analytic and combinatorial properties (most of which are at present only conjectural) of the entire function
F(x,y) = Σn≥0 xn/n! yn(n-1)/2
.
This function (or formal power series) arises in numerous
problems in enumerative combinatorics, notably in the
enumeration of connected graphs, and in statistical mechanics
in connection with the Potts model on the complete graph
(``mean-field'' or Curie--Weiss Potts model). This circle of problems also touches on the theory of integrable systems in classical mechanics (Calogero--Moser system). If time permits I will discuss an analogous problem for the "partial theta function":
Θ0(x,y) = Σn≥0 xn yn(n-1)/2
in this case some striking results can be proven, by using identities for q-series. For details, see http://www.maths.qmul.ac.uk/~pjc/csgnotes/sokal/
and http://arxiv.org/abs/1106.1003.
Warren Weaver Hall Room 517 at 10:00 am.
Friday, November 11 - Two consecutive talks.
Subhrosekhar Ghosh, UC Berkeley
What does a Point Process Outside a Domain tell us about What's Inside?
Abstract:
In a Poisson point process we have independence between disjoint spatial domains, so the points outside a disk give us no information on the points inside. The story gets a lot more interesting for processes with stronger spatial correlation. In the case of Ginibre ensemble, a process arising from eigenvalues of random matrices, we prove that the outside points determine exactly the number of points inside, and further, we demonstrate that they determine nothing more. In the case of zero ensembles of Gaussian power series, we prove that the outside points determine exactly the number and the centre of mass of the inside points, and nothing further. These phenomena suggest a certain hierarchy of point processes based on their rigidity; Poisson, Ginibre and the Gaussian power series fit in at levels 0, 1 and 2 in this ladder.
Time permitting, we will also look at some interesting consequences of our results, with applications to continuum percolation, reconstruction of Gaussian entire functions, and others. This is based on joint work with Fedor Nazarov, Yuval Peres and Mikhail Sodin.
Warren Weaver Hall Room 317 at 10:00 am.
Barry McCoy, State University of New York, Stony Brook
The Romance of the Ising model
Abstract:
The essence of romance is mystery. In this talk I will explore the meaning of this for the Ising model, beginning in 1946 with Bruria Kaufman and Willis Lamb, to the wedding of the Ising model with Painlevé functions, to the discovery of a possible natural boundary in the susceptibility and concluding with recent work (and mysteries) on the factorization of the form factor expansion and the relation of the diagonal susceptibility to p+1Fp hypergeometric functions, modular forms and particular Calabi-Yau equations.
Warren Weaver Hall Room 317 at 11:05 am.
Thursday and Friday, November 17-18
Friday, November 25
Thanksgiving holiday. No Seminar.
Friday, December 2 - Two consecutive talks.
Leonid Koralov, University of Maryland
Polymer measures and branching diffusions.
Abstract:
We study two problems related by a common
set of techniques. In the first problem, we consider
a model for the distribution of a long homopolymer
in a potential field. For various values of the temperature, including
those at or near the critical value, we consider the limiting behavior of
the polymer when its size tends to infinity.
In the second problem, we investigate the long-time evolution
of branching diffusion processes in inhomogeneous media.
The qualitative behavior of the processes
depends on the intensity of the branching.
In the super-critical case, we describe the asymptotics of the number
of particles in a given domain and describe the growth of
the region containing the particles. In the sub-critical regime,
we describe the limiting distribution of the total number of particles.
Warren Weaver Hall Room 317 at 10:00 am.
Michael Damron, Princeton University
A simplified proof of the relation between scaling exponents in
first-passage percolation.
Abstract:
In first passage percolation, we place i.i.d. non-negative
weights on the nearest-neighbor edges of Z^d and study the induced
random metric. A long-standing conjecture gives a relation between two
"scaling exponents": one describes the variance of the distance
between two points and the other describes the transversal
fluctuations of optimizing paths between the same points. This is
sometimes referred to as the "KPZ relation." In a recent breakthrough
work, Sourav Chatterjee proved this conjecture using a strong
definition of the exponents. I will discuss work I just completed with
Tuca Auffinger, in which we introduce a new and intuitive idea that
replaces Chatterjee's main argument and gives an alternative proof of
the relation. One advantage of our argument is that it does not
require a certain non-trivial technical assumption of Chatterjee on
the weight distribution.
Warren Weaver Hall Room 317 at 11:05 am.
Friday, December 9 - Two consecutive talks.
Charles Radin, University of Texas
Phase transitions in complex networks.
Abstract:
We consider the competition between structures in large
simple graphs, for instance the competition between the density of
edges and the density of triangles. Using "graph limits" to control the
asymptotics of probability distributions on graphs, one finds well
defined phases in the parameter space, with perfectly sharp
transitions, in close analogy with the liquid/gas and fluid/solid
transitions of statistical mechanics.
Warren Weaver Hall Room 317 at 10:00 am.
Yuri Kifer, Hebrew University
A Zoo of Nonconventional Limit Problems.
Abstract:
We discuss various limit theorems for "nonconventional" sums of the form Σ1≤n≤N B(ξ(q1(n)), ξ(q2(n)), ..., ξ(qℓ(n))) where ξ(n), n ≥ 0 is either a Markov chain or a hyperbolic (expanding, subshift of finite type etc.) transformation (i.e. then ξ(n) = Tnx) while qi(n), i ≤ k are linear and qj(n), k < j ≤ ℓ grow faster than linearly. The motivation for this study comes, in particular, from many papers about nonconventional ergodic theorems appeared in the last 30 years. Among our results are central limit theorem, large deviations, averaging and Poisson type limit theorems. We will talk also about some "nonconventional" multifractal formalism type problems computing the Hausdorff dimension of sets of numbers whose expansions have prescribed frequencies of combinations of digits in places qj(n), j = 1, 2, ..., ℓ; n ≥ 1.
Warren Weaver Hall Room 317 at 11:05 am.
Spring Semester 2011
Friday, January 28
Pieter Trapman, Stockholm University
Long-range percolation on the hierarchical lattice.
Abstract:
The hierarchical lattice of order N, may be seen as the leaves of an infinite regular N-tree, in which the distance between two vertices is the distance to their most recent common ancestor in the tree.
We create a random graph by independent long-range percolation on the hierarchical lattice of order N: The probability that a pair of vertices at (hierarchical) distance R share an edge depends only on R and is exponentially decaying in R, furthermore the presence or absence of different edges are independent.
We give criteria for percolation (the presence of an infinite cluster) and we show that in the supercritical parameter domain, the infinite component is unique. Furthermore, we show the percolation probability
(the density of the infinite cluster) is continuous in the model parameters. In particular, there is no percolation at criticality.
Joint work with Slavik Koval and Ronald Meester
Warren Weaver Hall Room 317 at 10:00 am.
Friday, February 4
S.R.S. Varadhan, Courant Institute, New York University
Large deviations for dense random graphs.
Abstract:
In this joint work with Sourav Chatterjee we investigate the large deviation properties of various subgraph counts in random graphs $G(n,p)$ having $n$ vertices with every unoriented edge having independently probability $p$ of being present. The large deviation is carried out in the space of "graph limits" with "cut topology" that allows for continuous contraction to subgraph counts. For example, questions like what is the most likely way the triangle count can be higher (or lower) by a factor from their expected values are answered and exhibit some qualitative changes in behavior as the parameters vary. Finally, there is a curious application to random matrices.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, February 11
Ioannis Karatzas, Columbia University
Stable diffusions interacting through their ranks, as models for large equity markets.
Abstract:
We introduce and study ergodic multidimensional diffusion processes interacting through their ranks. These interactions give rise to invariant measures which are in broad agreement with stability properties observed in large equity markets over long time-periods.
The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset. Such models are able realistically to capture critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study.
The methodologies used in this study touch upon the question of triple points for systems of interacting diffusions; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of collision local times.
The theory we develop has connections with the analysis of Queueing Networks in heavy traffic, as well as with models of competing particle systems in Statistical Mechanics such as the Sherrington-Kirkpatrick model for spin-glasses.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, February 18
Jason Miller, Microsoft Research
CLE(4) and the Gaussian Free Field.
Abstract:
The discrete Gaussian free field (DGFF) is the Gaussian measure on real-valued functions h(.) on a bounded subset D of the two dimensional integer lattice, whose covariance is given by the Green's function for simple random walk. The graph of h(.) is a random surface which serves as a physical model for an effective interface. We show that the collection of random loops given by the level sets of the DGFF at any height converges in the fine-mesh scaling limit to a family of loops which is invariant under conformal transformations when D is a lattice approximation of a non-trivial simply connected domain. In particular, there exists λ>0 such that the level sets whose height is an odd integer multiple of lambda converges to a nested conformal loop ensemble with parameter κ=4 (so-called CLE(4)), a conformally invariant measure on loops which locally look like SLE(4). Using this result, we give a coupling of the continuum Gaussian free field (GFF), the fine-mesh scaling limit of the DGFF, and CLE(4) such that the GFF can be realized as a functional of CLE(4) and conversely CLE(4) can be made sense as a functional of the GFF. Based on joint work with Scott Sheffield.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, February 25
Will Perkins, Courant Institute, New York University
The Bohman-Frieze Process.
Abstract:
The Bohman-Frieze process is a simple modification of the Erdős-Rényi random graph that adds dependence between the edges biased in favor of joining isolated vertices. We present new results on the phase transition of the Bohman-Frieze process and show that qualitatively it belong to the same class as the Erdős-Rényi process. The results include the size and structure of small components in the barely sub- and supercritical time periods. We will also mention a class of random graph processes that seems to exhibit markedly different critical behavior.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, March 4 - Two consecutive talks.
Jian Ding, UC Berkeley
Cover times, blanket times, and the Gaussian free field.
Abstract:
The cover time of a finite graph (the expected time for the simple random
walk to visit all the vertices) has been extensively studied, yet a number
of fundamental questions concerning cover times have remained open.
Aldous and Fill (1994) asked whether there is a deterministic
polynomial-time algorithm that computes the cover time up to an O(1)
factor. Winkler and Zuckerman (1996) defined the blanket time (when the
empirical distribution is within a factor of 2, say, of the stationary
distribution) and conjectured that the blanket time is always within O(1)
of the cover time. The best approximation factor found so far for both
these problems was (log log n)^2 for n-vertex graphs, due to Kahn, Kim,
Lovasz, and Vu (2000).
We show that the cover time of a graph, appropriately normalized, is
proportional to the expected maximum of the (discrete) Gaussian free field
on G. We use this connection and Talagrand's majorizing measures theory to
deduce a positive answer to the question of Aldous and Fill and to
establish the conjecture of Winkler and Zuckerman. These results extend to
arbitrary reversible finite Markov chains. No prior knowledge of
Talagrand's theory or of cover times will be assumed. This is joint work
with James Lee and Yuval Peres.
Warren Weaver Hall Room 317 at 10:00 am.
Ivan Corwin, Courant Institute
Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions.
Abstract:
We consider the solution of the stochastic heat equation with multiplicative noise and delta function initial condition whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We prove explicit formulas for the one-dimensional marginal distributions -- the crossover distributions -- which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit. The limit also describes the crossover behaviour between the symmetric and asymmetric exclusion processes.
Warren Weaver Hall Room 317 at 11:15 am.
Friday, March 11
Asaf Nachmias , MIT
The phase transition in percolation on the Hamming cube.
Abstract:
Consider percolation on the Hamming cube {0,1}^n at the critical
probability p_c (at which the expected cluster size is 2^{n/3}). It is
known that if p=p_c(1+O(2^{-n/3}), then the largest component is of
size roughly 2^{2n/3} with high probability and that this quantity is
non-concentrated. We show that for any sequence eps(n) such that eps(n)>>2^{-n/3} and
eps(n)=o(1) percolation at p_c(1+eps(n)) yields a largest cluster of
size (2+o(1))eps(n)2^n.
This result settles a conjecture of Borgs, Chayes, van der Hofstad,
Slade and Spencer.
Joint work with Remco van der Hofstad.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, March 18
Spring recess. No Seminar.
Friday, March 25
Tonći Antunović, UC Berkeley
Some path properties of Brownian motion with variable drift.
Abstract:
If B is a Brownian motion and f is a function in the Dirichlet space, then by Cameron-Martin theorem, the process (B - f) has the same almost sure path properties as B. In this talk we will present some properties of the image and the zero set of Brownian motion perturbed by certain less regular drifts f (examples include Hilbert curves and Cantor functions). Based on joint works with Krzysztof Burdzy, Yuval Peres, Julia Ruscher and Brigitta Vermesi.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, April 1
Antonio Auffinger, Courant Institute
Heavy Tailed Random Matrices and Directed Polymers.
Abstract:
The sum of iid random variables properly scaled does not always converge
to a Gaussian distribution as in the CLT. If they are heavy tailed the
scaling constant changes and the limit law is no longer Gaussian.
In this talk I will show analogous results in three different models where
different and new limiting processes arise: the largest eigenvalue of
random matrices, the last passage time in last passage percolation and the
path measure in Directed Polymers in Random Environments. The main goal is
to compare the domain of attraction of (conjectured/proved) universality
phenomena of these models. This talk is based on joint works with G. Ben Arous (Courant) and O.
Louidor (UCLA) and S.Péché (Grenoble).
Warren Weaver Hall Room 317 at 10:00 am.
Friday, April 8
Allan Sly, Microsoft Research
Phase transitions and the complexity of counting.
Abstract:
Phase transitions have been conjectured to determine the computational complexity of a number of natural combinatorial counting problems. In this talk I will discuss the discrete hardcore model and its relationship to counting the independent sets of a graph.
We show that unless NP=RP there is no polynomial time approximation scheme for the partition function of the hardcore model (a weighted sum of independent sets) on graphs of maximum degree d for fugacity \lambda_c<\lambda<\lambda_c + \epsilon(d) where \lambda_c is the uniqueness threshold on the d-regular tree. Weitz produced an efficient algorithm for approximating the partition function when 0 < \lambda < \lambda_c(d) so this result demonstrates that the computational threshold exactly coincides with the statistical physics phase transition thus confirming a conjecture of Mossel, Weitz and Wormald.
The proof hinges on a detailed understanding of the distribution of the hardcore model on random bi-partite graphs using the small graph conditioning theorem from combinatorics and point to set correlations of extremal Gibbs measures.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, April 15
Columbia-Princeton Probability Day
Schedule (tentative)
| 09:00-10:00 AM | Registration and continental breakfast |
| 10:00-11:00 AM | Alexei Borodin |
| 11:00-12:00 AM | Shige Peng |
| 12:00-01:30 PM | Lunch |
| 01:30-02:30 PM | Yakov Sinai |
| 02:30-03:30 PM | Horng-Tzer Yau |
| 03:30-04:00 PM | Coffee break |
| 04:00-04:25 PM | Antonio Auffinger |
| 04:25-04:50 PM | Ilya Vinogradov |
| 04:50-05:15 PM | Hana Kogan |
Robertson Hall, Room 001 on the Princeton University campus. Full program and map here.
Friday, April 22
Patricia Gonçalves, University of Minho.
Scaling limits of additive functionals of exclusion processes.
Abstract:
In this talk I will consider exclusion processes denoted by (ηt)t≥0, evolving on ℤ and starting from the invariant state: the Bernoulli product measure (νρ)ρ∈[0,1]. The goal of the talk consists in establishing scaling limits of the functional
Γt(f) := ∫[0,t]f(ηs) ds
for proper local functions f. When f(η) := η(x), the functional Γt(f) is called the occupation time of the origin. I will present a method that was recently introduced in Goncalves and Jara (10') "Universality of the KPZ equation", from which we derive a local Boltzmann-Gibbs Principle for a class of exclusion processes. For the occupation time of the origin, this principle says that the functional Γt(f) is very well approximated to the density of particles. As a consequence, the scaling limits of Γt(f) follow from the scaling limits of the density of particles. As examples I will present the symmetric simple exclusion, the mean-zero exclusion and the weakly asymmetric simple exclusion. For the latter, when the asymmetry is strong enough such that the fluctuations of the density of particles are given by the KPZ equation, we establish the limit of Γt(f) in terms of this solution. The case of asymmetric simple exclusion will also be discussed.
This is a joint work with Milton Jara (IMPA-Brazil).
Warren Weaver Hall Room 317 at 10:00 am.
Friday, April 29
Jiří Černý, ETH, Zürich.
Vacant set of random walk on (random) graphs.
Abstract:
The vacant set is the set of vertices not visited by a random
walk on a graph before a given time T. In the talk, I will discuss
properties of this random subset of the graph, the phase transition
conjectured in its connectivity properties (in the `thermodynamic limit'
when |G| and T grow simultaneously), and the relation of the problem to the
random interlacement percolation. I will then concentrate on the case when
G is a large-girth expander or a random regular graph, where the
conjectured phase transition (and much more) can be proved.
Warren Weaver Hall Room 317 at 10:00 am.
Monday, May 2 - Special Seminar (Note the time and place)
Josef Teichmann, ETH, Zürich.
Affine Processes on Positive Semi-Definite Matrices
Abstract:
Classification and applications of affine processes on
positive semi-definite matrices is presented. These processes contain
OU processes on positive semi-definite matrices, and Wishart
processes. Generalizations towards symmetric cones are discussed.
Warren Weaver Hall Room 1314 at 10:00 am.
Friday, May 6 - No talk today
Friday, June 3
Mark Holmes,University of Auckland.
Random walks in degenerate random environments.
Abstract:
In joint work with Tom Salisbury, we study random walks in i.i.d. random environments in Z^2 in dimensions 2 and higher. In our environments, at any given site some steps may not be available to the random walker (i.e. we don't assume ellipticity). Among our main results are 0-1 laws for directional transience (extending results already known under the assumption of ellipticity) and a simple monotonicity result for 2-valued environments (at each site the environment takes one of two values).
Warren Weaver Hall Room 317 at 10:00 am.
Fall Semester 2010
Friday, September 24
Partha Dey, Courant Institute
Central Limit Theorem for First-Passage Percolation across thin cylinders.
Abstract:
We consider first-passage percolation on the graph ℤ×{-hn,-hn+1,...,hn}d-1 where each edge has an i.i.d. nonnegative weight. The passage time for a path is defined as the sum of weights of all the edges in that path and the first-passage time between two vertices is defined as the minimum passage time over all paths joining the two vertices. We show that the first-passage time Tn between the origin and the vertex (n,0,...,0) satisfies a Gaussian CLT as long as hn=o(nα) with α < 1/(d+1). The proof is based on moment estimates, a decomposition of Tn as an approximate sum of independent random variables and a renormalization type argument. We conjecture that the CLT holds upto hn=o(n2/3) for d=2 and provide some numerical support for that.
Joint work with Sourav Chatterjee.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, October 1
Ellen Saada, Université Paris 5
Euler hydrodynamics for attractive particle systems in random environment.
Abstract:
We prove quenched hydrodynamic limit under hyperbolic time scaling for bounded attractive particle systems on ℤ in random ergodic environment. Our result is a strong law of large numbers.
Joint work with C. Bahadoran, H. Guiol, K. Ravishankar.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, October 8
Tom LaGatta, Courant Institute
Riemannian First-Passage Percolation
Abstract:
Riemannian first-passage percolation is a continuum analogue of lattice FPP. Instead of considering a random metric on the lattice ℤ2, we begin with a random Riemannian metric on ℝ2. The global structures of the two models are similar - with my advisor Janek Wehr, we have proved a shape theorem for this model, which shows that balls under the Riemannian metric grow asymptotically like Euclidean balls. However, there is also a rich local structure, since Riemannian geometry provides us with notions of curvature and geodesics, curves which (locally) minimize length. Geodesics need not always globally minimize length (e.g., great circles on the sphere), and it is an interesting and important question to identify those geodesics which do so. No geometric background will be required for this talk.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, October 15
Clément Hongler, Columbia University
Conformal invariance of the Ising energy field.
Abstract:
We consider the planar Ising model from a conformal invariance point of view. We are interested in the scaling limit of the model at criticality. Physics theories, notably Conformal Field Theory, predict the existence of two conformal fields underlying the model: the spin and the energy density. We have recently proved the conjectured formulae for the energy field, with an improved precision, using discrete complex analysis techniques, thanks to the introduction of holomorphic spinors. More specifically, we relate the correlation functions of the energy to special values of the spinors, and prove convergence of the latter to continuous holomorphic spinors, giving scaling formulae for the correlation functions.
Partly based on joint work with Stas Smirnov.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, October 22 - Two consecutive talks.
Christophe Bahadoran, Université Blaise Pascal
Quasi-potential for the asymmetric exclusion process.
Abstract:
The purpose of this work is to recover by a dynamical approach the stationary large deviation functional derived by Derrida, Lebowitz & Speer (2003) for the asymmetric exclusion process in contact with reservoirs. A remarkable feature of this functional is its nonlocality, which is a signature of long-range correlations. The DLS functional is recovered and somewhat generalized by computing the quasi-potential associated to a suitable dynamical energy functional. While this approach was set up by Bertini et al. (2002) for symmetric and weakly asymmetric systems, it was so far lacking for strongly asymmetric systems, due to the different nature of the dynamical functional. The latter is a combination of a bulk functional based on entropy production (Jensen 2000, Varadhan 2004, Belletini et al. 2010) and boundary costs that measure violation from Bardos-Leroux-Nédélec boundary conditions in Burgers's equation (Bodineau & Derrida 2005).
Warren Weaver Hall Room 317 at 10:00 am.
Domokos Szász, Budapest University of Technology
Energy transfer and joint diffusion.
Abstract:
The joint diffusion of two particles in a dynamical environment was shown to become asymptotically independent for a 1-D degenerate mechanical model (Harris-Spitzer model) by the speaker in 1980, and for stochastic models of symmetric exclusion by Kipnis and Varadhan in 1985. In truly mechanical systems, however, where the interaction of the particles also involves energy exchange, this independence does not hold anymore. The phenomenon is explained and demonstrated for a stochastic model of two Lorentz disks. The diffusive limit of the motion of one particle is a mixture of Wiener processes and the random covariances are determined by the Boltzmann's Stosszahlansatz. The results are joint with Zs. Pajor-Gyulai.
Warren Weaver Hall Room 317 at 11:15 am.
Friday, October 29
Fredrik Johansson Viklund, Columbia University
Convergence rates for loop-erased random walk
Abstract:
Loop-erased random walk (LERW) is a self-avoiding random walk obtained by chronologically erasing the loops of a simple random walk. In the
plane, the lattice size scaling limit of LERW is known to be SLE(2), a random fractal curve constructed by solving the Loewner differential
equation with a Brownian motion input.
In the talk, we will discuss recent joint work with C. Benes (CUNY) and M. Kozdron (U. of Regina) on obtaining a rate for the convergence
of LERW to SLE(2). More precisely, we will outline our derivation of a rate for the convergence of the Loewner driving function for LERW to
Brownian motion with speed 2 on the unit circle, the Loewner driving function for SLE(2).
We will then show how to use this to obtain a rate for the convergence of the paths with respect to Hausdorff distance. Time permitting, we
will also indicate how some of these results can be extended to certain other models known to converge to SLE.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, November 5
Louis-Pierre Arguin, Courant Institute
Statistics of Branching Brownian Motion at the edge
Abstract:
Branching Brownian motion (BBM) is a Markov process where particles perform Brownian motion and independently split into two independent Brownian particles after an exponential holding time. The extreme value statistics of BBM in the limit of large time is of interest since BBM constitutes a borderline case, among Gaussian processes, where correlations start to affect the statistics. The law of the maximum of BBM has been understood since the works of Bramson and McKean. But little is known about the distribution of the particles close to the maximum. In this talk, I will present results on the correlation structure of these particles. This is used to unravel a Poissonian structure underlying the point process of particles at the edge. This is joint work with A. Bovier and N. Kistler.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, November 12 - Two consecutive talks.
Horng-Tzer Yau, Harvard University.
Random matrices and the conjectures of Wigner and Dyson.
Abstract:
Random matrices were introduced by E. Wigner to model the excitation spectra of large nuclei. The central idea is based on the hypothesis that the local statistics of the excitation spectrum for a large complicated system is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. Dyson Brownian motion is the flow of eigenvalues of random matrices when each matrix element performs independent Brownian motions. In this lecture, we will explain the connection between the universality of random matrices and the approach to local equilibrium of Dyson Brownian motion. This connection has led to a complete solution of the universality conjecture by Wigner, Dyson, Gaudin and Mehta.
The main tools in our approach are an estimate on the flow of entropy in Dyson Brownian motion and a local semicircle law. One key feature of the entropy estimate is an extension of the logarithmic Sobolev inequality to cases not covered by the convexity criterion of Bakry and Emery.
Warren Weaver Hall Room 317 at 10:00 am.
Tom Alberts, University of Toronto.
Intermediate Disorder for Directed Polymers in Dimension 1+1, and the Continuum Random Polymer.
Abstract:
The 1+1 dimensional directed polymer model is a Gibbs measure on simple random walk paths of a prescribed length. The weights for the measure are determined by a random environment occupying space-time lattice sites, and the measure favors paths to which the environment gives high energy. For each inverse temperature $\beta$ the polymer is said to be in the weak disorder regime if the environment has little effect on it, and the strong disorder regime otherwise. In dimension 1+1 it turns out that all positive $\beta$ are in the strong disorder regime. I will introduce a new regime called intermediate disorder, which is accessed by scaling the inverse temperature to zero with the length $n$ of the polymer. The precise scaling is $\beta n^{-1/4}$. The most interesting result is that under this scaling the polymer has diffusive fluctuations, but the fluctuations themselves are not Gaussian. Instead they are still coupled to the random environment, and their distribution is intimately related to the Tracy-Widom distribution for the largest eigenvalue of a random matrix from the GUE. More recent work also indicates that we can take a scaling limit of the entire intermediate disorder regime to construct a continuous random path under the effect of a continuum random environment. We call the scaling limit the continuum random polymer. I will discuss a few properties of the continuum random polymer and its intimate connection to the stochastic heat equation in one dimension.
Joint work with Kostya Khanin and Jeremy Quastel.
Warren Weaver Hall Room 317 at 11:15 am.
Friday, November 26
Thanksgiving holiday. No Seminar.
Friday, December 3
Shirshendu Chatterjee, Cornell University
Asymptotic Behavior of Aldous' Gossip Process.
Abstract:
Aldous (2007) defined a gossip process in which space is a discrete torus of size N, and the state of the process at time t is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each and at rate N^{-\alpha} to a site chosen at random from the torus. We will be interested in the case in which α < 3, where the long range transmission significantly accelerates the time at which everyone knows the information. We prove three results that precisely describe the spread of information and asymptotic behavior of the cover time in a slightly simplified model on the (real) torus.
Warren Weaver Hall Room 317 at 10:00 am.
Friday, December 10 - Two consecutive talks.
Michael Aizenman, Princeton University
Resonant Delocalization through Large Deviations for Random Operators on Tree Graphs.
Abstract:
We resolve an existing question concerning the mobility edge for operators with a hopping term and a random potential on regular tree graphs. The model has been among the earliest studied for Anderson localization, and it continues to attract attention because of analogies with localization issues for many particle systems. A resonance mechanism is identified which causes the somewhat surprising appearance of absolutely continuous spectrum well beyond the energy band of the operator's hopping term. For weak disorder this includes a Lifshitz tail regime of very low density of states.
(Joint work with S. Warzel.)
Warren Weaver Hall Room 317 at 10:00 am.
Janek Wehr, University of Arizona
Brownian motion in a diffusion gradient and exotic stochastic integrals.
Abstract:
A Brownian particle with a diffusion coefficient varying in space obeys a Newton equation of motion with a stochastic term. In the Smoluchowski-Kramers (or: overdamped) approximation, the mass of the particle is formally put equal zero, yielding a first order stochastic differential equation, which admits different interpretations, depending on the definition of the stochastic integral adopted. A recent experiment shows that the correct interpretation is "backwards Ito". I will show how this can be derived from taking the zero mass limit carefully and then discuss a whole class of similar problems. The overdamped limits can lead to equations with any definition of stochastic integration, including Ito, and backwards Ito and Stratonovitch (as a limiting case). Moreover, in a majority of these equations, the stochastic integral convention changes depending on the state of the system, going beyond Ito, Stratonovitch or any other standard definition. A series of experiments with electric circuits designed to verify these predictions is in its initial phase. This is a joint work with an experimental group in Stuttgart and with Scott Hottovy, a graduate student at the University of Arizona.
Warren Weaver Hall Room 317 at 11:15 am.
Spring Semester 2010
Friday, January 22
Pierre Nolin, Courant Institute
Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model
Warren Weaver Hall Room 101 at 10:00 am.
Tuesday, January 26
Special Seminar
ClÈment Hongler, UniversitÈ de GenËve
"Convergence of Ising model interfaces to dipolar SLE"
Warren Weaver Hall Room 1314 at 5:00 pm.
NOTE the date and room change.
Friday, January 29
Antonio Auffinger, Courant Institute
Random Matrices and Complexity of Spin Glasses
Warren Weaver Hall Room 317 at 10:00 am.
Friday, February 5
Ivan Corwin, Courant Institute
Fluctuations of the totally asymmetric simple exclusion process
Warren Weaver Hall Room 317 at 10:00 am.
Friday, February 12
Pierluigi Falco, Institute of Advanced Study
Rigorous evaluation of critical exponents through scaling limit
Warren Weaver Hall Room 317 at 10:00 am.
Friday, February 19
Antti Kemppainen, UniversitÈ Paris-Sud and University of Helsinki
Random curves, scaling limits and Loewner evolutions
Warren Weaver Hall Room 317 at 10:00 am.
Friday, February 26
Rama Cont, Columbia University
Functional Itô calculus, integration by parts and stochastic integral representation of martingale functionals
Warren Weaver Hall Room 317 at 10:00 am.
Friday, March 5
Van Vu, Rutgers
Random matrices: Universality of the Local eigenvalues statistics
Warren Weaver Hall Room 317,
at 2 pm.
Friday, March 12
Krishnamurthi Ravishankar, SUNY
Marking the Brownian web and applications
Warren Weaver Hall Room 317 at 10:00 am.
Friday, March 19
Spring Break - No Seminar.
Friday, March 26
Eyal Lubetzky, Microsoft Research
Critical slowdown for the Ising model on the two-dimensional lattice
Warren Weaver Hall Room 317 at 10:00 am.
Friday, April 2
Two Talks:
Geoffrey Grimmett, University of Cambridge
Embeddings, entanglement, and percolation
(Warren Weaver Hall Room 317 at 10:00am)
Chiranjib Mukherjee, Max Planck Institute
Brownian intersection local times and large deviations
(second talk will follow first)
Tuesday, April 6
Special Seminar
G·bor Pete, University of Toronto
"Random walk on percolation clusters, and scale-invariant groups"
Room 1302, at 10:00am.
NOTE the date and room change.
Friday, April 9
S. R. Srinivasa Varadhan, Courant Institute
On some central limit theorems by Martingale Approximation
Warren Weaver Hall Room 317 at 10:00 am.
Friday, April 16
Sinan G¸nt¸rk, Courant Institute
Quantization of Random Linear Measurements
Warren Weaver Hall Room 317 at 10:00 am.
Friday, April 23
Two Talks:
Govind Menon, Brown University
Lax equations and kinetic theory for shock clustering and Burgers turbulence
Warren Weaver Hall Room 317 at 10:00 am.
Eric Nordenstam, UniversitÈ Catholique de Louvain
A particle dynamics related to the shuffling algorithm for the Aztec diamond
Seminar cancelled due to Eyjafjallajˆkull
Friday, April 30
Two Talks:
Ori Gurel-Gurevich, Microsoft Research
Poisson Thickening
Warren Weaver Hall Room 317 at 10:00 am.
Sandy Zabell, Northwestern University
A large deviation result for pinned random walks with barrier curves
Warren Weaver Hall Room 605 at 12:30 pm.
Friday, May 7
Scott Sheffield, MIT
Internal DLA and the Gaussian free field
Warren Weaver Hall Room 317 at 10:00 am.
Friday, May 14
Eric Nordenstam, UniversitÈ Catholique de Louvain
A particle dynamics related to the shuffling algorithm for the Aztec diamond
Warren Weaver Hall Room 317 at 10:00 am.
Fall Semester 2009
Wednesdays and Fridays, September 9-30
Minerva Research Foundation Lectures at Columbia University
Jean Bertoin, Universite Paris VI
Exchangeable Coalescents
Wednesdays 10am-noon (Math 622) and Fridays 10am-noon (Math 507)
2990 Broadway at 117'th st., Columbia University
More information at the
announcement.
Friday, September 11
Mihyun Kang, Technische Universit‰t Berlin
Two critical behaviour of random planar graphs
Warren Weaver Hall Room 412 at 10:10 am.
Applied Mathematics Seminar talk
Patrick Dondl, Bonn University
Pinning of interfaces in random media
Warren Weaver Hall Rm 1302 at 2:30pm.
More information at the
Applied Mathematics Seminar website.
Friday, September 18
Paul Bourgade, TÈlÈcom-ParisTech
Random matrices on compact groups and independence
Warren Weaver Hall Room 517 at 12:00 noon.
Friday, September 25
Olivier Bernardi, MIT
Bijective approach to tree-rooted maps
Warren Weaver Hall Room 102 at 10:00 am.
Friday, October 2
Sourav Chatterjee, NYU and UC Berkeley
Superconcentration
Warren Weaver Hall Room 517 at 10:00 am.
Friday, October 9
Francesco Russo, INRIA Rocquencourt, Projet MATHFI and UniversitÈ Paris 13
Probabilistic representation of a generalized porous media type equation and related fields
Warren Weaver Hall Room 517 at 10:00 am.
Friday, October 16
Jian Ding, University of California, Berkeley
Near-critical random graph: its structure, diameter and mixing time
Warren Weaver Hall Room 517 at 10:00 am.
Informal Lunchtime Seminar (bring your lunch!)
Mark Meckes, Case Western Reserve University
Concentration of polynomials in random matrices
Warren Weaver Hall Room 1314 at 12:10 - 1:10pm.
Friday, October 23
Alexander Fribergh, NYU
Biased random walks on a percolation cluster
Warren Weaver Hall Room 517 at 10:00 am.
Friday, October 30
Ron Peled, NYU
High-dimensional homomorphism height functions are flat
Warren Weaver Hall Room 517 at 10:00 am.
Friday, November 6
Two Seminars
Oren Louidor, NYU
Finite connections for supercritical Bernoulli bond percolation in 2D.
Partha Dey, University of California, Berkeley
Stein's method and large deviation for number of triangles in ErdösñRÈnyi Random Graph
Both seminars are at Warren Weaver Hall Room 517, starting at 10:00 am.
Friday, November 13
Mark Kelbert, Swansea University
The branching diffusion on hyperbolic space
Warren Weaver Hall Room 517 at 10:00 am.
Thursday and Friday, November 19-20
Eighth Northeast Probability Seminar (NEPS)
Invited Speakers:
Rick Kenyon, Brown University
Claudia Neuhauser, University of Minnesota
Giovanni Peccati, UniversitÈ Paris Ouest
Craig Tracy, University of California, Davis
Held at the C.P. Davis Auditorium in the Schapiro Center at Columbia University.
More details at the seminar website.
Friday, November 27
Thanksgiving Holiday - No Seminar.
Friday, December 4
Lorenzo Zambotti, UniversitÈ Paris VI
An entropic functional on families of random variables from theoretical biology
Warren Weaver Hall Room 517 at 10:00 am.
Special Seminar
Fredrik Johansson, KTH
Behavior of the SLE path at the tip
Warren Weaver Hall Room 517 at 2:00 pm.
Friday, December 11
Rob van den Berg, Vrije Universiteit and CWI
Sharpness of percolation transitions in some dependent two-dimensional models
Warren Weaver Hall Room 517 at 10:00 am.
Spring Semester 2009
Friday, January 30
No seminar scheduled to allow people to hear Andrei Okounkov speak on "Random surfaces and Algebraic curves" at Columbia.
Lecture is at 9:30am, 520 Math, Columbia university.
Okounkov's talk will be followed by Thierry Bodineau's talk on "Large deviations for non-equilibrium particle systems" in Columbia's probability seminar.
12:00 noon, 903 SSW Bldg (1255 Amsterdam Avenue-btwn. 121st & 122nd Street).
Friday, February 6
Todd Kemp, MIT
"Resolvents of $R$-Diagonal Ensembles"
Warren Weaver Hall Room 312 at 10:00 am.
Friday, February 13
Michael Damron, Courant Institute
"Two Dimensional Invasion Percolation and Incipient Infinite Clusters"
Michael was ill and instead Ron Peled from Courant Insitute talked about "Translation-equivariant colorings of Poisson-Voronoi diagrams".
Warren Weaver Hall Room 312 at 10:00am.
Friday, February 20
Sasha Sodin, Tel-Aviv University
"Universality at the edge of the spectrum for random matrices with independent entries: Soshnikov's theorems and some extensions"
Warren Weaver Hall Room 312 at 10:00 am.
Monday, February 23 and Wednesday, February 25
Special lectures in probability at Columbia University
Etienne Pardoux, Marseille
First talk is on Monday, February 23, 9:30-11:00am, Hamilton 517, Columbia University.
"Can a single mutant's progeny survive for ever?"
Second talk is on Wednesday, February 25, 9:30-11:00am, SSW 1025, Columbia University.
"'Homegenization and SPDE's"
Note the unusual place and time! More details at the Columbia Probability Seminar website.
Friday, February 27
Dmitry Panchenko, Texas A&M
"The Ghirlanda-Guerra identities and ultrametricity in the Sherrington-Kirkpatrick model"
(Warren Weaver Hall Room 312 at 10:00am)
Friday, March 6
Special Joint Columbia / Courant Seminar
Yuval Peres, Microsoft Research
"Is the critical percolation probability local?"
(Warren Weaver Hall Room 312 at 10:00am)
Note also Yuval Peres' talk at the Courant Institute Mathematics Colloquium on Monday, March 2'nd.
Friday, March 13
Vincent Vargas, Université Paris Dauphine
"Stochastic scale invariance and KPZ equation"
(Warren Weaver Hall Room 312 at 10:00am)
Friday, March 20
Spring Break - No seminar scheduled
Friday, March 27
Kay Kirkpatrick, MIT
"Quantum many-body systems and the nonlinear Schroedinger equation"
(Warren Weaver Hall Room 312 at 10:00am)
Friday, April 3
Martin Hairer, Courant Institute
"A 'weak convergence' alternative to Harris chains"
(Warren Weaver Hall Room 312 at 10:00am)
Friday, April 10
Lionel Levine, MIT
"Growth rates and explosions in sandpiles"
(Warren Weaver Hall Room 312 at 10:00am)
Friday, April 17
Two Talks:
Xue-Mei Li, University of Warwick
"A negative result for Stochastic Differential Equations"
(Warren Weaver Hall Room 312 at 10:00am)
Charles Radin, University of Texas at Austin
"Modeling Sand"
(second talk will follow first)
Friday, April 24
Seminar Cancelled.
Monday, April 27
Special Seminar
Federico Camia, Vrije Universiteit
"Ising(Conformal) Fields and Cluster Area Measures"
Warren Weaver Hall Room 1302, at 10:00am.
NOTE the date and room change.
Wednesday, April 29
Special Seminar
Horng-Tzer Yau, Harvard University
"Dyson's Sine Kernel, Wigner Random Matrices, and Interacting Particle Systems"
Warren Weaver Hall Room 512 at 11:30am.
NOTE the date and room change.
Friday, May 1
Columbia-Princeton Probability Day
Held at Columbia University, Schermerhorn Hall room 501.
Registration/coffee from 9-10am, lectures follow.
Full program and map here.
The NYU spring semester ends on Monday, May 4'th. Some talks will be held during the summer, see below.
Friday, May 15
Yuri Kifer, Hebrew University of Jerusalem
"Nonconventional Limit Theorems"
(Warren Weaver Hall Room 312 at 10:00am)
Thursday, July 2
ClÈment Hongler, UniversitÈ de GenËve
The energy density in the 2D Ising model
Warren Weaver Hall Room 1314 at 11:00am.
NOTE the date and room change.
Fall Semester 2008
Friday, November 7
Thierry Bodineau, École Normale Supérieure
"Current large deviations in stochastic systems"
(Warren Weaver Hall Room 312 at 10:00 am)
Friday, October 24
Joint Columbia/Courant seminar (note the unusual place and time)
Brian Rider, University of Colorado, Boulder
"Beta Ensembles, Random Schroedinger, and Diffusion"
(Room 507
at Columbia at 10:45 am)
George Papanicolaou, Stanford University
"Modeling fine-scale uncertainty in Bayesian parameter estimation and applications"
(Room 520
at Columbia at 12:00 pm)
Friday, October 17
Michael Damron, Courant Institute
"Invasion percolation in 2D"
(Warren Weaver Hall Room 312 at 10:00 am)
Friday, October 10
José Ramírez, Universidad de Costa Rica
"Diffusion limits for eigenvalues of random matrices"
(Warren Weaver Hall Room 312 at 10:00 am)
Friday, October 3
Pierre Nolin, Courant Institute
"A particular bit of universality: inhomogeneity and SLE(6)"
(Warren Weaver Hall Room 312 at 10:00 am)
Friday, September 26
Ofer Zeitouni, Weizmann Institute & University of Minnesota
"Exit measures for isotropic Random walk in random environments - a perturbative approach"
(Warren Weaver Hall Room 312 at 10:00 am)
Nina Gantert, Universität Münster
"Survival time of random walk in random environment among soft obstacles"
(Warren Weaver Hall Room 312 at 11:15 am)
Abstracts
Spring 2010
Pierre Nolin, Courant Institute
Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model .
Abstract: For two-dimensional independent percolation, Russo-Seymour-Welsh (RSW) bounds on crossing probabilities are an important a-priori indication of scale invariance, and they turned out to be instrumental to describe the phase transition. They are in particular a key tool to derive the so-called scaling relations, that link the critical exponents associated with the main macroscopic functions.
In this talk, we prove RSW-type uniform bounds on crossing probabilities for the FK Ising model at criticality, independent of the boundary conditions. A central tool in our proof is Smirnov's fermionic observable for the FK Ising model, that makes appear some harmonicity on the discrete level, providing precise estimates on boundary connection probabilities. We also prove several related results - some new and some not - among which the fact that there is no magnetization at criticality, tightness properties for the interfaces, and the value of the half-plane one-arm exponent.
This is joint work with H. Duminil-Copin and C. Hongler.
ClÈment Hongler, UniversitÈ de GenËve
Convergence of Ising model interfaces to dipolar SLE.
Abstract: We consider the interfaces of the critical planar Ising model on the square lattice. In a Dobrushin setup, that is, when the boundary conditions are [+] on a boundary arc and [-] on the rest, the interface between [+] and [-] spins has been shown by Smirnov (and Chelkak-Smirnov for more general lattices) to converge to chordal SLE(3).
The three types of boundary conditions of the Ising model that have been predicted to be conformally invariant are [+], [-] and free, as well as combinations of them. In the case of dipolar boundary conditions, that is, when the boundary is split into [+], [-] and free arcs, the interface starting between [+] and [-] has been conjecture by physicists to converge to a variant of SLE, called dipolar SLE(3), thus generalizing Smirnov's result.
We will give the proof of this conjecture. It relies on the introduction of a new martingale observable, which plays the role of a stochastically conserved quantity, and helps deducing conformal symmetry of the model.
The introduction of the martingale observable is made through the introduction of a dual model. Using a remarkable combinatorial identity, known as Kramers-Wannier duality, we prove that obtaining a martingale observable for the interface can be made by understanding spin-spin correlations on a dual Ising model.
Using the FK representation of this dual Ising model and the scaling limit of its interfaces which is SLE(16/3), as well as the convergence of the discrete holomorphic fermions introduced by Smirnov for the FK and the spin Ising models, we manage to express these spin-spin correlations as SLE integrals. These integrals are finally computed using Conformal Field Theory-inspired computations (relying notably on solutions of Dotsenko-Fateev equations).
Our method is in fairly general and allows in principle to identify the scaling limit of interfaces in all the conformally invariant boundary conditions setups. It can moreover be used to prove early predictions about crossing probabilities in the Ising model and is the starting point of the construction of a free boundary conditions version of the Conformal Loop Ensembles.
Antonio Auffinger, Courant Institute
Random Matrices and Complexity of Spin Glasses.
Abstract: We introduce a new identity, relating random
matrix theory and the problem of counting the number of critical points of certain random
(Gaussian) functions in high dimensional spheres, the Hamiltonians of spherical spin-glass models.
This identity allows us to describe an interesting layered structure of local minima and saddle points at low levels of energy and
to compute the ground state energy of these Hamiltonians.
This is joint work with G. Ben Arous (Courant) and J. Cerny (ETHZ).
Ivan Corwin, Courant Institute
Fluctuations of the totally asymmetric simple exclusion process.
Abstract: We study how the evolution of this process fluctuates around its expected behavior. For TASEP started with two-sided Bernoulli initial conditions we provide a complete characterization of the limiting (large time) fluctuation processes. These processes vary according to the region in the hydrodynamic limit. Results proved can be interpreted also in terms of last passage percolation, crystal growth models, queues in series, and spiked Wishart random matrices.
This includes joint with GÈrard Ben Arous, and with Patrik Ferrari and Sandrine PÈchÈ.
Pierluigi Falco, Institute of Advanced Study
Rigorous evaluation of critical exponents through scaling limit.
Abstract: I will introduce some critical features of the Eight-Vertex and the Ashkin-Teller models; and I will discuss how the use of
Renormalization Group permits the rigorous proof of some scaling
formulas conjectured by Kadanoff. In the Eight-Vertex case, these
formulas give the exact values of some critical exponents that are not
computable through the Baxter's exact solution.
Antti Kemppainen, UniversitÈ Paris-Sud and University of Helsinki
Random curves, scaling limits and Loewner evolutions.
Abstract: In the 2D statistical physics and its lattice models, interfaces are random curves. A general method to prove the convergence of a random discrete curve, as the lattice mesh goes to zero, is to first show the existence of subsequent scaling limits and then to prove the uniqueness. In this talk, I will introduce a sufficient condition, and some equivalent formulations, that guarantee the precompactness (existence) and also that the limits are Loewner evolutions, i.e. they correspond to continuous Loewner driving processes. The second result is needed for the unique characterization of the limits. This framework of estimates can be used for almost all of the already existing proofs of an interface converging to a Schramm-Loewner evolution (SLE), and for at least one new result. In principle, it can be applied beyond SLE.
Joint work with Stanislav Smirnov, UniversitÈ de GenËve
Rama Cont, Columbia University
Functional Itô calculus, integration by parts and stochastic integral representation of martingale functionals.
Abstract: We develop a non-anticipative calculus for functionals of a
continuous semimartingale, using a pathwise functional
derivative recently proposed by B Dupire. A functional extension of
the Ito formula is derived, and used to obtain a constructive
martingale representation theorem for a class of continuous
martingales verifying a regularity property.
By contrast with the Clark-Ocone formula, this representation involves
non-anticipative quantities which may
be computed pathwise. The martingale representation formula allows to
obtain an integration by parts formula for Ito stochastic
integrals, which in turn enables to define a non-anticipative weak
functional derivative for a class of square-integrable martingales.
We show that this weak derivative is the adjoint of Ito stochastic
integral and may be viewed as a non-anticipative ``lifting" of the
Malliavin derivative. Finally, regular functionals of an Ito
martingale which have the local
martingale property are characterized as solutions of a functional
differential equation, for which a uniqueness result is given.
Joint work with: David FOURNIE (Columbia University).
Van Vu, Rutgers
Random matrices: Universality of the Local eigenvalues statistics.
Abstract: One of the main goals of the theory of random matrices is to establish the limiting distributions of the eigenvalues. In the 1950s, Wigner proved his famous semi-cirle law (subsequently extended by Anord, Pastur and others), which established the global distribution of the eigenvalues of random Hermitian matrices.
In the last fifty years or so, the focus of the theory has been on the local distributions, such as the distribution of the gaps between consecutive eigenvalues, the k-point correlations, the local fluctuation of a particular eigenvalue, or the distribution of the least singular value. Many of these problems have connections to other fields of mathematics, such as combinatorics, number theory, statistics and numerical linear algebra.
Most of the local statistics can be computed explicitly for random matrices with gaussian entries (GUE or GOE), thanks to Ginibre's formulae of the joint density of eigenvalues. It has been conjectured that in the limit the same results hold for other models of random matrices. This is generally known as the Universality phenomenon, which has been supported by overwhelming numerical evidence
and various concrete conjectures.
In this talk, we would like to discuss recent progresses concerning the Universality phenomenon, focusing on a recent result (obtained jointly with T. Tao), which asserts that all local statistics of eigenvalues of a random matrix are determined by the first four moments of the entries.
This provides the answer to several old problems.
The method also extends to other models of random matrices, such as sample covariance matrices.
Krishnamurthi Ravishankar, SUNY
Marking the Brownian web and applications
Abstract
Eyal Lubetzky, Microsoft Research
Critical slowdown for the Ising model on the two-dimensional lattice.
Abstract: Intensive study throughout the last three decades has yielded a rigorous
understanding of the spectral-gap of the Glauber dynamics for the Ising
model on $Z^2$ everywhere except at criticality. While the critical behavior
of the Ising model has long been the focus for physicists, mathematicians
have only recently developed an understanding of its critical geometry with
the advent of SLE, CLE and new tools to study conformally invariant systems.
A rich interplay exists between the static and dynamic models. At the static
phase-transition for Ising, the dynamics is conjectured to undergo a
critical slowdown: At high temperature the inverse-gap is $O(1)$, at the
critical $\beta_c$ it is polynomial in the side-length and at low
temperature it is exponential in it. A long series of papers verified this
on $Z^2$ except at $\beta=\beta_c$ where the behavior remained unknown.
In this work we establish the first rigorous polynomial upper bound for the
critical mixing, thus confirming the critical slowdown for the Ising model
in $Z^2$. Namely, we show that on a finite box with arbitrary (e.g. fixed,
free, periodic) boundary conditions, the inverse-gap at $\beta=\beta_c$ is
polynomial in the side-length. The proof harnesses recent understanding of
the scaling limit of critical Fortuin-Kasteleyn representation of the Ising
model together with classical tools from the analysis of Markov chains.
Based on joint work with Allan Sly.
Geoffrey Grimmett, University of Cambridge
Embeddings, entanglement, and percolation.
Abstract: Can there exist a monotone embedding of one infinite random word inside another, with bounded gaps? What can be said about the critical point for the existence of an infinite `entangled' set of open edges in the percolation model on the cubic lattice?
These two questions are connected through a new type of percolation process, called `Lipschitz percolation'. It will be shown (amongst other things) how to embed some higher-dimensional words, and to obtain the best (so far) lower bound for the entanglement critical point.
This work is joint with Ander Holroyd, and has benefited from the Courant Institute seminar series.
Chiranjib Mukherjee, Max Planck Institute
Brownian intersection local times and large deviations.
Abstract: We consider a number of independent Brownian motions running in the d-dimensional Euclidean space until they exit a fixed ball. We look
at the spatial intersection of the paths. Le Gall and others constructed
an object measuring the intensity of the path intersections in the set.
Keeping track of the notion of local time pertaining to a single path,
this object is called the ``Brownian intersection local time''. Koenig and
Moerters recently studied the upper tails of this random object, sending
the amount of intersection on a fixed compact set to infinity. The
resulting variational formula admits minimizer(s) with certain
probabilistic interpretation along the line of classical Donsker-Varadhan
theory. Inspired by this, we study large deviations for normalised
intersection local times (as a measure) for a fixed time horizon in the
ball. As a corollary to this, we obtain an LDP for normalised intersection
local times, for motions observed until individual exit times.
G·bor Pete, University of Toronto
Random walk on percolation clusters, and scale-invariant groups.
Abstract: There are well-known connections between geometric properties of
Cayley graphs and the behavior of simple random walk on them. But most
tools stop working if we consider random walk inside an infinite
percolation cluster of the graph, even though the same results should
hold.
In the first part of the talk, I give a *simple* proof that the
isoperimetric profile of the infinite cluster basically coincides with the
profile of the lattice Z^d for any percolation density p>p_c(Z^d), and for
p close enough to 1 on Cayley graphs of finitely presented groups. This
implies that the on-diagonal heat kernel decay survives percolation.
The situation on Z^d is better than in general because of a standard
percolation technique called renormalization. So, in the second part, I
will examine the possibility of renormalization on other Cayley graphs. A
group G is called scale-invariant if it has a nested chain of finite index
subgroups, all isomorphic to G, whose intersection is trivial. Itai
Benjamini conjectured that scale-invariant groups must have polynomial
volume growth. In joint work with V. Nekrashevych, I have given several
counterexamples, including the lamplighter group Z_2 \wr Z.
I will give a lot of open questions.
S. R. Srinivasa Varadhan, Courant Institute
On some central limit theorems by Martingale Approximation.
Abstract: We will investigate the CLT for sums of the form $\sum_{i=1}^n f(X_i, X_{2i},\ldots, X_{ki})$ where $\{X_i\}$ are dependent random variables
with some mixing properties.
Sinan G¸nt¸rk, Courant Institute
Quantization of Random Linear Measurements.
Abstract: We will discuss the problem of how to quantize m random linear measurements of k-dimensional vectors, where m > k. The standard choice is to round each measurement vector to the nearest lattice point, and to reconstruct via the (Moore-Penrose) pseudo-inverse. This talk is about a quantization and reconstruction alternative which relies on the concept of "noise-shaping" in analog-to-digital conversion, Sobolev-dual frames, and concentration of singular values for certain families of random matrices. We will also present implications and improvements for compressed sensing.
Joint work with M. Lammers, A. Powell, R. Saab, and O. Yilmaz.
Govind Menon, Brown University
Lax equations and kinetic theory for shock clustering and Burgers turbulence.
Abstract: Much of our current understanding of statistical theories of turbulence relies on vastly simplified caricatures. One such caricature is Burgers turbulence. This is the study of the statistics of shocks in Burgers equation with random initial data or forcing. This model also arises in statistics, combinatorics, and models of coagulation and surface growth. It is of wide interest as a benchmark, even if it describes phenomena that are not entirely turbulent.
I will describe a kinetic theory for shock clustering that applies to all scalar conservation laws with convex flux. A remarkable feature of the kinetic theory is that it is presented as a Lax pair, admits surprising exact solutions, and has intriguinging connections with completely integrable systems and random matrix theory. This is mostly joint work with Bob Pego (CMU) and Ravi Srinivasan (UT, Austin).
Eric Nordenstam, UniversitÈ Catholique de Louvain
A particle dynamics related to the shuffling algorithm for the Aztec diamond.
Abstract: The shuffling algorithm (introduced by Elkies et al.) for sampling a
tiling of the Aztec diamond uniformly at random can be seen as a certain
dynamics on a set of interacting particles. This is a discretisation of
a model of interlaced Brownian motions recently studied by Warren. As
an application of these results, I will sketch a new proof of that fact
that, in a suitable scaling limit of large Aztec diamonds, one can
recover the distribution of the eigenvalues of a GUE matrix and its
principal minors.
This work is related to recent work of Borodin and Ferrari.
Ori Gurel-Gurevich, Microsoft Research
Poisson Thickening.
Abstract: Can a Poisson process be thickened?
That is, can more points be added deterministically to a Poisson process,
so that the resulting process is also a Poisson process (of higher intensity)?
We will show that this can be done, but not equivariantly
(i.e. not in a way which commutes with some shift).
In recent years, there has been much interest in problems of this kind:
given a stochastic spatial process X, can it be extended to another process Y
(perhaps under additional constraints)? For example, can the cells of a
Poisson-Voronoi tessellation be colored deterministically and equivariantly,
such that adjacent cells have different colors?
We will survey results of this kind, with particular emphasis
on those which yield pretty pictures and explain the solution to the thickening
problem in some detail.
Joint Work with Ron Peled.
Sandy Zabell, Northwestern University
A large deviation result for pinned random walks with barrier curves
Abstract
Scott Sheffield, MIT
Internal DLA and the Gaussian free field.
Abstract: Internal diffusion limited aggregation (DLA) is a simple and natural random growth model with a beautiful history. I will describe some recent work joint with David Jerison and Lionel Levine on this subject. This work includes a proof of the "logarithmic-fluctuation" conjecture. It also precisely describes the scaling limit of the random fluctuations. The Gaussian free field makes a surprise appearance.
Fall 2009
Mihyun Kang, Technische Universit‰t Berlin
Two critical behaviour of random planar graphs
Abstract
Paul Bourgade, TÈlÈcom-ParisTech
Random matrices on compact groups and independence.
Abstract: The Chinese restaurant process gives an iterative construction of the
Ewens measures on the symmetric group. We will apply this idea to any
unitary group, generating in particular its Haar measure by composing
independent reflections. As a consequence, for a random matrix uniformly
distributed on a compact group, the characteristic polynomial is a product
of independent random variables. We will also explain how these results
are linked to classical number-theoretic conjectures.
Olivier Bernardi, MIT
Bijective approach to tree-rooted maps
Abstract: Planar maps are connected planar graphs embedded in the 2-dimensional sphere, and considered up to homeomorphisms. These objects are of interest, in particular, as models of random geometries. Many recent advances in the theory of maps are based on bijections between maps and certain decorated plane trees.
In this talk, I will consider "tree-rooted maps", that is, maps with a marked spanning tree. I will present a bijection between tree-rooted maps and pairs of plane trees. I will explain the link between this bijection and a bijection by Schaeffer/Bouttier-Di Francesco-Guitter which is fundamental for studying the metric properties of maps. Lastly, I will present a generalization of the bijection to orientable surfaces other than the sphere and its enumerative consequences.
Sourav Chatterjee, NYU and UC Berkeley
Superconcentration
Abstract: We introduce the term `superconcentration' to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens, the field becomes chaotic under small perturbations and a `multiple valley picture' emerges. Conversely, chaos implies superconcentration. While a few notable examples of superconcentrated functions already exist, e.g. the largest eigenvalue of a GUE matrix, we show that the phenomenon is widespread in physical models; for example, superconcentration is present in the Sherrington-Kirkpatrick model of spin glasses, directed polymers in random environment, the Gaussian free field and the Kauffman-Levin model of evolutionary biology. As a consequence we resolve the long-standing physics conjectures of disorder-chaos and multiple valleys in the Sherrington-Kirkpatrick model, which is one of the focal points of this talk.
Francesco Russo, INRIA Rocquencourt, Projet MATHFI and UniversitÈ Paris 13
Probabilistic representation of a generalized porous media type equation and related fields
Abstract
Jian Ding, University of California, Berkeley
Near-critical random graph: its structure, diameter and mixing time
Abstract
Mark Meckes, Case Western Reserve University
Concentration of polynomials in random matrices
Abstract:In the spirit of results of Guionnet and Zeitouni and of free probability theory, we prove concentration inequalities for noncommutative polynomials of large independent random matrices. This is joint work with S. Szarek.
Alexander Fribergh, NYU
Biased random walks on a percolation cluster
Abstract:We will present a model of random walk in random environments (RWRE) called biased random walks on a percolation cluster. This model arises from the physics literature and exhibits an unexpected slowdown phenomenon, the asymptotic speed of the random walk may actually decrease as the bias is increased. We will describe this phenomenon, how it arises and describe many open questions related to it. We will then explain how one can understand the speed of the walk on a percolation cluster of high density (p close to 1).
Ron Peled, NYU
High-dimensional homomorphism height functions are flat
Abstract:A homomorphism height function on a graph G is an integer-valued function on the vertices of G which differs by exactly one across every edge of G. One is concerned with the properties of the typical height function, that is, a function sampled uniformly among all height functions which equal 0 at some fixed point. This is a generalization of simple random walk - the case when G is a path. We take G to be a d-dimensional torus. In this case, height functions correspond to proper 3-colorings, at least for certain boundary conditions. Our main result is that in high enough dimensions, the typical height function is very flat, having bounded height at any fixed vertex and small global fluctuations. Indeed, we obtain a structure theorem for the typical function showing that it is almost constant on either the even or odd sublattices of the torus, with precise estimates for the size of breakups of this pattern. This extends results of Kahn and Galvin for the case that G is the hypercube.
Using an observation of Yadin, the results extend also to the related class of 1-Lipschitz functions on G. In addition, some information is provided on the two dimensional torus case hinting that it undergoes a certain roughening transition. This refutes a conjecture of Benjamini, Yadin and Yehudayoff.
Oren Louidor, NYU
Finite connections for supercritical Bernoulli bond percolation in 2D
Abstract:Two vertices are said to be finitely connected if they belong to the same cluster and this cluster is finite. We derive sharp asymptotics for finite connection probabilities for supercritical Bernoulli bond percolation on Z^2.
Partha Dey, University of California, Berkeley
Stein's method and large deviation for number of triangles in
ErdösñRÈnyi model Random Graph
Abstract:Stein's method is a semi-classical tool for establishing
distributional convergence, particularly effective in problems
involving complex dependencies. A general way of deriving
concentration inequalities using Stein's method was introduced by
Sourav Chatterjee in 2005. Here we show how this method can be used to
derive exact large deviation asymptotics for the number of triangles
in the ErdösñRÈnyi Random Graph G(n,p) when p>=0.31. The
proof is based on a rigorous analysis of the exponential random graph
model using Stein's method for exchangeable pair. The same idea can be
extended to find large deviation rate function for number of small
subgraphs in G(n,p) for p above a threshold. This talk is based on
joint work with Sourav Chatterjee.
Mark Kelbert, Swansea University
The branching diffusion on hyperbolic space
Abstract: We say that a branching diffusion (BD) on a Riemannian manifold $M$ is recurrent if at least one offspring of a single particle starting from
$x\in M$ will return to any neighborhood of point $x$ with probability 1,
and transient otherwise. The sufficient conditions for recurrency and
transiency of BD are presented. For a transient BD on a hyperbolic space
with a variable fission rate the Hausdorff dimension of the attractor on
the absolute is evaluated.
Lorenzo Zambotti, UniversitÈ Paris VI
An entropic functional on families of random variables from theoretical biology
Abstract: G. Edelman, O. Sporns and G. Tononi have introduced in theoretical biology the neural complexity of a family of random variables, defining it as a specific average of mutual information over subsystems. We provide a mathematical framework for this concept, studying
in particular the problem of maximization of such functionals for fixed system
size and the asymptotic properties of maximizers as the system size goes to infinity.
(Joint work with Jerome Buzzi)
Fredrik Johansson, KTH
Behavior of the SLE path at the tip
Abstract: The Schramm-Loewner evolution (SLE) is a family of random fractal curves that arise as scaling limits of two-dimensional lattice models from statistical physics. In the talk I will discuss a derivation of the optimal Holder exponent for the SLE path (in the capacity parameterization) and, briefly, a related result on the decay of harmonic measure at the tip. This is joint work with G. Lawler (University of Chicago).
Rob van den Berg, Vrije Universiteit and CWI
Sharpness of percolation transitions in some dependent two-dimensional models
Abstract: Ordinary (independent) percolation models have a sharp percolation transition: below the percolation threshold the cluster size distribution has exponential decay. For 2-dimensional models this was first proved by Kesten (1980).
In 1981 Russo proved a so-called approximate zero-one law and pointed out that a key step in Kesten's argument can be seen as a special case of this more general law. A few years ago, new results by Bollobas and Riordan for the two-dimensional Voroinoi percolation model triggered more research in that direction.
I will mainly focus on the contact process, a mathematical model of spatial epidemics, vegetation patterns and other natural random spatial structures.
Spring 2009
Todd Kemp, MIT
Resolvents of $R$-Diagonal Ensembles
Abstract: Random matrix theory, a very young subject, studies the behaviour of the eigenvalues of matrices with random entries (with specified correlations). When all entries are independent (the simplest interesting assumption), a universal law emerges: essentially regardless of the laws of the entries, the eigenvalues become uniformly distributed in the unit disc as the matrix size increases. This
circular law was first proved, with strong assumptions, in the 1980s; the current state of the art, due to Tao and Vu, with very weak assumptions, is less than a year old. It is the
universality of the law that is of key interest.
What if the entries are
not independent? Of course, much more complex behaviour is possible in general. In the 1990s, "$R$-diagonal" matrix ensembles were introduced; they form a large class of non-normal random matrices with (typically) non-independent entries. In the last decade, they have found many uses in operator theory and free probability; most notably, they feature prominently in Haagerup's recent work towards proving the invariant subspace conjecture.
In this lecture, I will discuss my recent joint work with Haagerup and Speicher, where we prove a universal law for the resolvent of any $R$-diagonal operator. The circular ensemble is an important special case. The rate of blow-up is, in fact, universal among
all $R$-diagonal operators, with a constant depending only on their fourth moment. The proof intertwines both complex analysis and combinatorics.
Michael Damron, Courant Institute
Two Dimensional Invasion Percolation and Incipient Infinite Clusters
Abstract: In this talk, we will examine the structure of the two dimensional invasion percolation cluster (IPC) of the origin. We will review recent results about the sizes of the ponds and talk about their relation to multiple-armed generalizations of Kesten's incipient infinite cluster (IIC). In addition we will give the ideas of the proof of mutual singularity of the IPC and IIC measures.
This is joint work with Artem Sapozhnikov and Balint Vagvolgyi.
Sasha Sodin, Tel-Aviv University
Universality at the edge of the spectrum for random matrices with independent entries: Soshnikov's theorems and some extensions
Abstract: We shall discuss the distribution of extreme eigenvalues
for several classes of random matrices with independent
entries. In particular, we shall discuss the results of
Soshnikov and some of their recent extensions, and the combinatorial questions that appear in the proofs.
Based on joint work with Ohad Feldheim.
Dmitry Panchenko, Texas A&M
The Ghirlanda-Guerra identities and ultrametricity in the Sherrington-Kirkpatrick model
Abstract: The Parisi theory of the SK model completely describes the geometry of the Gibbs sample in a sense that it predicts the limiting joint distribution of all scalar products, or overlaps, between i.i.d.
replicas. One of the main properties of this distribution is the ultrametricity which means that the Gibbs
measure approximately concentrates on the ultrametric subset of all configurations; another property
is the Ghirlanda-Guerra distributional identities. It is well known that these two properties completely
determine the distribution and, probably for this reason, they were always considered complementary.
We show that if in the limit an overlap takes finitely many values then the Ghirlanda-Guerra identities
actually imply ultrametricity.
Yuval Peres, Microsoft Research
Is the critical percolation probability local?
Abstract: We show that the critical probability for percolation on a d-regular non-amenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. This is a special case of a conjecture due to O. Schramm on the locality of p_c. We also prove a finite analogue of the conjecture for expander graphs.
Joint work with Itai Benjamini and Asaf Nachmias.
Vincent Vargas, Université Paris Dauphine
Stochastic Scale Invariance and the KPZ formula
Abstract: In this talk, we will prove the KPZ equation (initially introduced in the
framework of quantum gravity in 2 dimensions) for the limit lognormal
measures introduced by Mandelbrot. More specifically, for a given set K,
we will relate it's Hausdorff dimension under the Euclidian metric to it's
Hausdorff dimension under the random metric induced by the limit lognormal
measure. We will see how the notion of stochastic scale invariance is
crucial in the derivation of the aforementioned relation. (Joint work with
R. Rhodes)
Kay Kirkpatrick, MIT
Quantum many-body systems and the nonlinear Schroedinger equation
Abstract: At extremely cold temperatures there forms a new state of matter, called Bose-Einstein condensation, with weird behavior: quantum effects are visible macroscopically, and friction no longer matters. Certain aspects of this phenomenon are nicely understood by scaling limits.
We describe two scaling limits for systems of many quantum particles: mean-field systems and Bose-Einstein condensation. First, in mean-field systems, the microscopic particles experience weak and diffuse interactions, and the Hartree equation provides the macroscopic description. Second, in Bose-Einstein condensation (which can be viewed as a limiting case of mean-field systems), the particles experience strong and short-scale interactions, and the cubic nonlinear Schroedinger equation provides the macroscopic description.
In recent joint work with Benjamin Schlein and Gigliola Staffilani, we have handled the two-dimensional Bose-Einstein condensation--and the periodic case is especially interesting, as it involves some techniques from analytic number theory.
Martin Hairer, Courant Institute
A 'weak convergence' alternative to Harris chains
Abstract: One of the most commonly used theories to prove (strong) convergence of a Markov
chain to its invariant measure is the theory of Harris chains. One major drawback
of this theory is that it requires a lower bound on transition probabilities, which fails
to hold in many infinite-dimensional examples where transition probabilities are
mutually singular. This is the case for example for some stochastic PDEs, as well as
some stochastic delay equations.
We provide an alternative theory which allows to obtain constructive criteria for weak
convergence, thus exploiting the topology of the state space. In particular, we obtain a
"weak form" of Harris's theorem, which yields spectral gap results in Wasserstein-type
distances. These results are also of interest in the finite-dimensional case as they yield
simple stability theorems for the invariant measure under weak approximations of the
semigroup.
Lionel Levine, MIT
Growth rates and explosions in sandpiles
Abstract: The abelian sandpile model in Z^d produces beautiful examples of
pattern formation, most of which are not yet well understood. I'll
discuss a pair of conjectures about the scale invariance and
dimensional reduction of the patterns formed. I'll also explore the
dichotomy between robust and explosive sandpile configurations. The
former are configurations to which adding a finite amount of
additional sand produces only finitely many topplings. An example is
the constant configuration of 2 chips at each site in Z^2. We prove a
"least action principle" and use it to bound the diameter of the set
of sites that topple. If an arbitrarily small fraction of sites
chosen at random start with 3 chips instead of 2, however, the result
is an explosion: every site in Z^2 topples infinitely often.
Joint work with Anne Fey and Yuval Peres.
Xue-Mei Li, University of Warwick
A negative result for Stochastic Differential Equations
Abstract: The solution to an ordinary differential equation depends on its
initial data continuously provided that it has a global solution.
This is not the case for stochastic differential equations. Positive
results have been searched for long and hard. For a global strong
solution to exist, the vector fields should have linear growth at infinity
(in the forward direction), allowing logarithmic order corrections.
The regularity needed for the vector fields are locally Lipschtz.
The question is how to construct examples of conservative SDEs
which has no global smooth solutions. The counter examples we
knew so far do not satisfy the linear grwoth condition.
We construct a SDE without a global smooth flow whose coefficients
are bounded and smooth. Only finite dimensional noise is needed. This is
joint work with M. Scheutzow.
Federico Camia, Vrije Universiteit
Ising(Conformal) Fields and Cluster Area Measures
Abstract: I will discuss a representation for the magnetization field of the
critical two-dimensional Ising model in the scaling limit as a random
field using renormalized area measures associated with SLE
clusters. The renormalized areas come from the scaling limit of critical
FK (Fortuin-Kasteleyn) clusters and the random field is a convergent sum
of the area measures with random signs. The representation is based on the
interpretation of the lattice magnetization as the sum of the signed areas
of clusters. If time permits, potential extensions, including to three
dimensions, will also be discussed. The talk will be based on joint work
with Chuck Newman and on work in progress
with Chuck Newman and C. Garban.
Horng-Tzer Yau, Harvard University
Dyson's Sine Kernel, Wigner Random Matrices, and Interacting Particle Systems
Abstract:The local eigenvalue statistics of the Gaussian Unitary Ensemble (GUE) is given by Dyson's Sine kernel.
It was conjectured that this law holds for a much general class of random matrices--- the universality
conjecture of random matrices. For matrix ensembles that are unitarily invariant, there has been a great progress
using technique from orthogonal polynomials. For the case of Hermitian Wigner random matrices
i.e. for matrix ensembles with i.i.d. entries are in general not unitarily invariant,
the only result is due to Johansson who proved the sine kernel for N by N matrices that are of the form $H + t V$
where $H$ is distributed according to a Wigner matrix ensemble and $V$ has the law of GUE. The parameter $t$ is
required to be of order one. Our main result states that the Dyson's sine kernel holds for $t \ge N^{-3/4}$
i.e. for Wigner matrices with a vanishing Gaussian perturbation. Our approach is based on technique from interacting particle systems and key technical inputs
are the local semi-circle law and level repulsion for Wigner random matrices. We remark that the universality conjecture for general Wigner matrices could
be deduced from the case $t \ll N^{-1}$ which is still an open problem.
ClÈment Hongler, UniversitÈ de GenËve
The energy density in the 2D Ising model
Abstract:We study the Ising model from a conformal invariance point of view using discrete complex analysis
methods. We are here interested in the scaling limit at critical temperature
of the two-dimensional Ising model in a simply
connected domain with boundary. In particular, we are interested in the effect
of the boundary with some conditions (+ or free) on local observables. In this talk we will be interested
in the behaviour of the so-called energy density field at the scaling limit, giving a rigourous exact derivation
of predictions obtained using Conformal Field Theory, exhibiting a nice
connection with hyperbolic geometry.
This derivation is made through the study of a so-called fermionic observable which is discrete holomorphic
in a particular sense and converges to a holomorphic function in the scaling limit.
This is joint work with Stanislav Smirnov.
Past years
Go back to the index