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Friday January 27, 2017
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Pierre Tarres
(Université Paris-Dauphine)
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Reinforced Random Walks and Statistical Physics
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We explain how the Edge-reinforced random walk, introduced by Coppersmith
and Diaconis in 1986, is related to several models in statistical physics, namely the
supersymmetric hyperbolic sigma model studied by Disertori, Spencer and Zirnbauer
(2010), the random Schrödinger operator and Dynkin's isomorphism.
These correspondences enable us to show recurrence/transience results on the Edge-reinforced
random walk, and they also allow us to provide insight into these models. This
work is joint with Christophe Sabot, and part of it is also in collaboration with Margherita
Disertori, Titus Lupu and Xiaolin Zeng.
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Friday February 10, 2017
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Mickey Salins (Boston University)
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Uniform large deviations principle for a general class of stochastic partial differential equations.
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A rare weather event is sometimes called a “100 year storm” if the event
is so unlikely that it happens on average only about once per century.
As this phrase suggests, there is a strong connection between the probabilities
of rare events and the time it takes those events to occur. The theory of large deviations
was developed in the 1960s by Varadhan, Freidlin, Wentzell and others to quantify both the
decay rates of probabilities of rare events for finite-dimensional stochastic differential equations
and the growth rates of the so-called exit times, the amount of time it takes for those events to occur.
The exit time problems require the large deviations principles to be uniform with respect to initial conditions
in bounded sets. Over the past few decades, researches have proven uniform large deviations principles
for many examples of stochastic partial differential equations, but the methods tend to be equation
specific and dependent on the chosen topology of the function space. In this talk, I demonstrate how to
use a weak convergence approach and the uniform Laplace principle to prove large deviations principles
that are uniform with respect to initial conditions in bounded sets. This is a needed improvement over
the previous formulations which only could be used to prove uniformity over compact sets. The method
works for a large class of semilinear
Banach-space-valued stochastic differential equations whose linear part generates a compact semigroup.
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Friday February 17, 2017
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10:30AM Guillaume Barraquand (Columbia)
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KPZ growth in a half space
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The asymptotic fluctuations of a large class of growth processes and one dimensional particle
systems are predicted to follow probability distributions from random matrix theory with 1/3 scaling exponents.
It is conjectured that the limit theorems are universal, in the sense that they do not depend on the microscopic
details of the model. However, the geometry and boundary conditions have an influence on the nature of limiting
statistics. In this talk we will explore the situation in a half space. We will see how the limiting fluctuations
depend on the distance to the boundary and the boundary condition, using the example of last passage percolation
in a half-quadrant. It turns out that the algebraic structure behind the integrability of last passage percolation
is also related to the stochastic six-vertex model in a half quadrant. Via scaling limits, this leads to a limit
theorem for the current in ASEP when particles are confined to the positive integers.
Joint works with Jinho Baik, Alexei Borodin, Ivan Corwin, Toufic Suidan and Michael Wheeler.
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11:30 Andrea Agazzi (Stanford and Université de Genève)
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Large Deviations Theory for Chemical Reactions Networks
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At the microscopic level, the dynamics of networks of chemical reactions can be modeled as jump Markov processes.
The rates of these processes are in general neither uniformly Lipschitz continuous nor bounded away from zero,
obstructing the straightforward application of large deviation theory to this framework.
We bypass these issues by respectively applying tools of Lyapunov stability theory and recent
results on interacting particle systems. This way, we characterize a class of processes obeying a
LDP in path space, and extend the latter to infinite time intervals through Wentzell-Freidlin (W-F) theory.
Finally, we provide natural sufficient topological conditions on the network of reactions for the applicability
of our LDP and W-F results. These conditions can be checked algorithmically.
This is joint work with Amir Dembo and Jean-Pierre Eckmann.
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Friday February 24, 2017
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10:15 Xinyi Li (U. Chicago)
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Percolative properties of Brownian interlacements and its vacant set
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In this talk, I will give a brief introduction to Brownian interlacements,
and investigate various percolative properties regarding this model. Roughly speaking,
Brownian interlacements can be described as a certain Poissonian cloud of doubly-infinite
continuous trajectories in the d-dimensional Euclidean space, d greater or equal to 3, with
the intensity measure governed by a level parameter. We are interested in both the interlacement
set, which is an enlargement ("the sausages") of the union of the trace in the aforementioned
cloud of trajectories, and the vacant set, which is the complement of the interlacement set.
I will talk about the following results: 1) The interlacement set is "well-connected", i.e.,
any two "sausages" in d-dimensional Brownian interlacements, can be connected via no more
than ceiling((d-4)/2) intermediate sausages almost surely. 2) The vacant set undergoes a
non-trivial percolation phase transition when the level parameter varies.
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11:15 Daniel Ahlberg (IMPA and Uppsala University)
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Random coalescing geodesics in first-passage percolation
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A random metric on Z^2 is obtained by assigning non-negative i.i.d. weights to the edges of the
nearest neighbour lattice. We shall discuss properties of geodesics in this metric.
We develop an ergodic theory for infinite geodesics via the study of what we shall call `random coalescing geodesics’.
Random coalescing geodesics have a range of nice properties. By showing that they are (in some sense) dense is
the space of geodesics, we may extrapolate these properties to all infinite geodesics. As an application of this theory
we answer a question posed by Benjamini, Kalai and Schramm in 2003, that has come to be known as the `midpoint problem’.
This is joint work with Chris Hoffman.
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Friday March 03, 2017
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Philippe Sosoe (Harvard)
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A Sharp Quasi-Invariance Result for Gaussian Measures under NLS with Quartic Dispersion
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We present a result showing that Gaussian measures on Sobolev spaces are left quasi-invariance by
a nonlinear Schroedinger equation on the torus: the statistical distribution of the solution is absolutely continuous with
respect to that of the initial data. The result is sharp in the sense that it extends to all Sobolev spaces where the equation
is well-posed in a reasonable sense. This is a probabilistic manifestation of the familiar competition
between nonlinearity and dispersion: without the dispersive term, the distribution of the solution of the corresponding ODE on H^S
is singular with respect to the initial data for any positive time.
Joint work with Tadahiro Oh and Nikolai Tzvetkov.
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Friday March 10, 2017
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Xiaoqin Guo (U. Purdue)
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Harnack inequality for a balanced random environment
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We consider a random walk in a balanced random environment
on $Z^d$ which is allowed to be non-elliptic.
This is a Markov chain generated by a discrete non-divergence form operator.
In this talk, assuming that the environment is iid and “genuinely d-dimensional”,
we will present a Harnack inequality for discrete harmonic functions of the corresponding operator.
The result is based on the analysis of the percolation structure of the (non-reversible) environment
and renormalization arguments.
Joint work with N. Berger, J.-D. Deuschel and M.Cohen.
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Friday March 17, 2017
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Spring recess
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tba
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Friday March 24, 2017
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10:15 Elliot Paquette (Ohio State University)
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Random perturbations of non-normal matrices
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Suppose one wants to calculate the eigenvalues of a large, non-normal matrix.
For example, consider the matrix which is 0 in most places except above the diagonal, where it is 1.
The eigenvalues of this matrix are all 0. Similarly, if one conjugates this matrix, in exact arithmetic
one would get all eigenvalues equal to 0. However, when one makes floating point errors, the eigenvalues
of this matrix are dramatically different. One can model these errors as performing a small, random perturbation
to the matrix. And, far from being random, the eigenvalues of this perturbed matrix nearly exactly equidistributed
on the unit circle. This talk will give a probabilistic explanation of why this happens and discuss the general
question: how does one predict the eigenvalues of a large, non-normal, randomly perturbed matrix?
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11:15 Tom Trogdon (UC Irvine)
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Numerical analysis of random matrices
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Numerical analysis and random matrix theory have long been coupled, going (at least)
back to the seminal work of Goldstine and von Neumann (1951) on the condition number of random matrices.
The works of Trotter (1984) and Silverstein (1985) incorporate numerical techniques to assist in the analysis
of random matrices. One can also consider the problem of computing distributions (i.e. Tracy-Widom)
from random matrix theory. In this talk, I will discuss different numerical analysis problems:
(1) sampling random matrices and (2) using them to analyze the halting time (or runtime) of numerical algorithms.
For the latter, I will focus primarily on recent proofs of universality for the (inverse) power method, the QR
algorithm and the Toda algorithm.
This is joint work with P. Deift, G. Menon, S. Olver and R. Rao.
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Friday March 31, 2017
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Columbia-Princeton Probability day
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Friday April 07, 2017
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Yan Fyodorov (King's College London)
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Exponential number of equilibria and depinning threshold for directed polymers in a random potential
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Using the Kac-Rice approach, we show that the mean number of all possible equilibria of an elastic line
(directed polymer), confined in a harmonic well and submitted to a quenched random Gaussian potential
grows exponentially with its length L. The growth rate is found to be directly related to the fluctuations
of the Lyapunov exponent of an associated Anderson localization problem of a 1-d Schroedinger equation in
a random potential. Eventually this rate controls the value of a threshold for the depinning transition
in presence of an applied force, and we provide an upper bound for the threshold.
The talk is based on joint results with Pierre Le Doussal, Alberto Rosso and Christophe Texier.
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Thursday April 13, 2017 (3:20pm in WWH 1302)
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Vadim Gorin (MIT)
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Macroscopic fluctuations of discrete particle systems
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This is the first of a series of five lectures, see here for a detailed schedule.
Many particle systems of 2d statistical mechanics, random matrix theory, and asymptotic representation theory exhibit similar macroscopic asymptotic behavior leading to smooth limit shapes and log-correlated Gaussian fields. I will present two recent approaches to such theorems based on discrete loop equations and on Schur generating functions. |
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Friday April 14, 2017
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10:15 AM Antonio Auffinger (Northwestern Univ.)
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The SK model is FRSB at zero temperature.
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In the late 70’s and early 80’s, Giorgio Parisi wrote a series of ground breaking papers introducing the idea of replica symmetry breaking. His powerful insight allowed him to predict a solution for the SK model. In this talk, I will explain some of Parisi’s predictions for the SK model and prove that at zero temperature the model is full-step replica symmetry breaking. More precisely, I will show that at zero temperature the functional order parameter of the mixed $p$-spin model contains infinitely many points in its support. I will also describe the implications of this result for the description of the energy landscape.
Based on joints works with Wei-Kuo Chen (Minnesota) and Qiang Zeng (Northwestern).
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11:15 AM Ilya Goldsheid (Queen Mary University)
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Lyapunov exponents of products of non-identically distributed independent matrices and transformations.
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The asymptotic behavior of products of independent identically distributed $m\times m$ random matrices is now relatively well understood (at least if $m$ is fixed).
A long standing natural problem is: what part (and under what conditions) of the corresponding theory can be extended to the case of products of non-identically distributed matrices and, more generally, transformations?
Perturbation theory is a very natural example of a situation where such a question arises.
In my talk, I'll try to answer this question.
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Wednesday April 19, 2017
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Courant/Columbia probability seminar
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Jean Bertoin, Zhou Fan, Cyril Labbé
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tba
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Friday April 28, 2017
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Victor Chernozhukov
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Central Limit Theorems and Bootstrap in High Dimensions
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This paper derives central limit and bootstrap theorems for probabilities
that sums of centered high-dimensional random vectors hit hyperrectangles and
sparsely convex sets. Specifically, we derive Gaussian and bootstrap
approximations for probabilities $\Pr(n^{-1/2}\sum_{i=1}^n X_i\in A)$ where
$X_1,\dots,X_n$ are independent random vectors in $\mathbb{R}^p$ and $A$ is a
hyperrectangle, or, more generally, a sparsely convex set, and show that the
approximation error converges to zero even if $p=p_n\to \infty$ as $n \to
\infty$ and $p \gg n$; in particular, $p$ can be as large as $O(e^{Cn^c})$ for
some constants $c,C > 0$. The result holds uniformly over all hyperrectangles, or
more generally, sparsely convex sets, and does not require any restriction on
the correlation structure among coordinates of $X_i$. Sparsely convex sets are
sets that can be represented as intersections of many convex sets whose
indicator functions depend only on a small subset of their arguments, with
hyperrectangles being a special case. |
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Friday May 05, 2017
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Antti Kupiainen
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Probabilistic Liouville Theory
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I will review a probabilistic approach to the Liouville Conformal Field Theory (LCFT).
This is a nongaussian perturbation of the Gausssian Free Field which conjecturally
describes the scaling limit of planar maps decorated with critical models of statistical
mechanics (such as the Ising model). The multipoint correlation functions of the Liouville random field
can be given in terms of the partition function of a thermal particle in a 2d Gaussian Free Field
random environment and bound by logarithmic potentials to the insertion points.
This approch sheds light to puzzling properties of LCFTand paves the way to a proof of the celebrated
Dorm-Otto-Zamolodchicov-Zamolodchicov formula for the three point function. This is joint work with
F.David, R.Rhodes and V.Vargas.
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Friday May 26, 2017
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Alan Hammond
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The weight, geometry and coalescence of scaled polymers in Brownian last passage percolation.
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In last passage percolation (LPP) models, a random environment in the two-dimensional integer lattice
consisting of independent and identically distributed weights is considered. The weight of an upright
path is said to be the sum of the weights encountered along the path. A principal object of study are the polymers, which are the upright paths whose weight is maximal given the two endpoints. Polymers move in straight lines over long distances with a two-thirds exponent dictating fluctuation. It is natural to seek to study collective polymer behaviour in scaled coordinates that take account of this linear behaviour and the two-third exponent-determined fluctuation.
We study Brownian LPP, a model whose integrable properties find an attractive probabilistic expression.
Building on a study ( arXiv:1609.02971 ) concerning the decay in probability for the existence of several near polymers
with common endpoints, we demonstrate that the probability that there exist k disjoint polymers across a unit box in
scaled coordinates has a superpolynomial decay rate in k.
This result has implications for the Brownian regularity of the scaled polymer weight profile begun
from rather general initial data.
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