Probability and Mathematical Physics Seminar
This seminar covers a wide range of topics in pure and applied probability and in mathematical physics. Unless otherwise noted, the talks take place on Fridays, 11:10am–12pm, in room WWH 1302. See directions to the Courant Institute.
To be kept informed, you can subscribe to the Probability Seminar Mailing list.
Twice every semester, we will have a joint Columbia–Courant meeting (hosted once at each institution) as the Probability and the City seminar.
Seminar Organizer(s): Eyal Lubetzky, Paul Bourgade, Hugo Falconet, and the probability group
Upcoming Events
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Friday, April 19, 202411:10AM, Warren Weaver Hall 1302
Non-mean-field random matrices related to quantum chaos and Anderson conjecture
Jun Yin, UCLASynopsis:
The Quantum Chaos Conjecture has long captivated the scientific community, proposing a crucial spectral phase transition demarcating integrable systems from chaotic systems in quantum mechanics. In integrable systems, eigenvectors typically exhibit localization with local eigenvalue statistics adhering to the Poisson distribution. In contrast, chaotic systems are characterized by delocalized eigenvectors, and their local eigenvalue statistics reflect the Sine kernel distribution reminiscent of the conventional random matrix ensembles GOE/GUE. Similarly, the Anderson conjecture reveals comparable phenomena in the context of disordered systems. This talk delves into the heart of this phenomenon, presenting a novel approach through the lens of random matrix models. By utilizing these models we aim to provide a clear and intuitive demonstration of the same phenomenon shedding light on the intricacies of these long-standing conjectures.
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Friday, April 26, 202411:10AM, Warren Weaver Hall 1302
Confinement of Unimodal Probability Distributions and an FKG-Gaussian Correlation Inequality
Mark Sellke, Harvard UniversityNotes:
While unimodal probability distributions are well understood in dimension 1, the same cannot be said in high dimension without imposing stronger conditions such as log-concavity. I will explain a new approach to proving confinement (e.g. variance upper bounds) for high-dimensional unimodal distributions which are not log-concave, based on an extension of Royen’s celebrated Gaussian correlation inequality. We will see how it yields new localization results for Ginzburg-Landau random surfaces with very general monotone potentials. Time permitting, I will also discuss a related result on the effective mass of the Fröhlich Polaron, which is joint work with Rodrigo Bazaes, Chiranjib Mukherjee, and S.R.S. Varadhan.
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Friday, May 3, 202411:10AM, Warren Weaver Hall 1302
TBA
Christopher Janjigian, Purdue University
Past Events
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Friday, April 12, 202411:10AM, Warren Weaver Hall 1302
Superdiffusion for Brownian motion with random drift
Ahmed Bou-Rabee, NYUSynopsis:
A Brownian particle subject to a random, divergence-free drift will have enhanced diffusion. The correlation structure of the drift determines the strength of the diffusion and there is a critical threshold, bordering the diffusive and superdiffusive regimes. Physicists have long expected logarithmic-type superdiffusivity at this threshold, and recently some progress in this direction has been made by mathematicians.
I will discuss joint work [arXiv:2404.01115] with Scott Armstrong and Tuomo Kuusi in which we identify and obtain the sharp rate of superdiffusivity. We also establish a quenched invariance principle under this scaling. Our proof is a quantitative renormalization group argument made rigorous by ideas from stochastic homogenization. -
Friday, March 29, 202411:10AM, Warren Weaver Hall 1302
The Critical 2d Stochastic Heat Flow and other critical SPDEs
Nikos Zygouras, University of WarwickSynopsis:
Thanks to the theories of Regularity Structures, Paracontrolled Distributions
and Renormalisation we now have a robust framework for singular SPDEs, which are “sub-critical”
in the sense of renormalisation. Recently, there have been efforts to approach the situation
of “critical” SPDEs and statistical mechanics models. A first such treatment has been
through the study of the two-dimensional stochastic heat equation, which has revealed a
certain phase transition and has led to the construction of the novel object called the
Critical 2d Stochastic Heat Flow. In this talk we will present some aspects of this model
and its construction. We will also present a variety of structures that underlie the critical dimension andsome further developments relating to other critical SPDEs.
Parts of this talk are based on joint works with Caravenna and Sun and others with Rosati and Gabriel. -
Friday, March 15, 20249:30AM, Columbia University, Mathematics Hall, 2990 Broadway 312
Workshop Universality and integrability in KPZ at Columbia University
Notes:
Workshop universality and integrability in KPZ (https://sites.google.com/view/universalityintegrabilityinkpz/home) organised at Columbia University from 03/11 to 03/15. Registration is required.
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Friday, March 8, 202410AM, Warren Weaver Hall 1302
Probability and the City seminar
Michael Aizenman (Princeton University), Daniel Stein (NYU) and Ivan Corwin (Columbia University)Notes:
Michael Aizenman (10am)
Title: Stochastic geometry of correlations in statistical and quantum mechanics
Abstract: The talk will start by recalling some of the pivotal contributions made by Charles M. Newman and collaborators in their rigorous analyses of critical phenomena in statistical mechanic, in which non-perturbative results of fundamental interest were derived through a broad range of analytic and probabilistic methods. Among those are techniques enabled by percolation-type stochastic geometric representations of the spread of correlations in systems of local interactions. The discussion will expand into such methods which, when available, are enabled by local conditional symmetries. Their more recent applications include proofs of discontinuity, or its absence, in symmetry breaking phase transitions in some paradigmatic classical and quantum spin models, and bounds on the quantum entanglement in the ground states of related quantum spin arrays.
(Results mentioned will include also speaker’s joint works with S. Warzel and H. Duminil-Copin).
Daniel Stein (11am)
Title: The Thermodynamic Structure of Short-Range Spin Glasses
Abstract: The thermodynamics of classical spin glasses, and more generally the statistical mechanics of quenched disorder, is a problem of general interest to physicists and mathematicians and one to which Chuck has made numerous contributions. Its importance was recognized in 2021 with the awarding of the Nobel Prize in Physics to Giorgio Parisi for his replica symmetry breaking solution of the Sherrington-Kirkpatrick model (a mean-field model of spin glasses) and its application to other problems in complex systems.I will begin with a brief review of the main features of replica symmetry breaking, including an application to physical systems. I will then turn to the problem of understanding the nature of the spin glass phase in nearest-neighbor classical spin glass models in finite dimensions. The central question to be addressed is the nature of broken symmetry in these systems. This is still a subject of controversy, and although the issues surrounding it have become more sharply defined in recent years, it remains an open question. I will explore this problem, introducing a mathematical construct called the metastate, to arrive at a unified picture of our current understanding of the structure of the spin glass phase in finite dimensions.
Ivan Corwin (12pm)Title: Scaling limit of colored ASEP.
Abstract: Each site x in Z is initially occupied by a particle of color -x. Across each bond (x,x+1) particles swap places at rate 1 or q<1 depending on whether they are in reverse order (e.g. color 2 then 1) or order (color 1 then 2). This process describes a bijection of Z-->Z which starts maximally in reverse order and randomly drifts towards being ordered. Another name for this model is the "colored asymmetric simple exclusion process". I will explain how to use the Yang-Baxter equation along with techniques involving Gibbs line ensemble to extract the space-time scaling limit of this process, as well as a discrete time analog, the "colored stochastic six vertex model". The limit is described by objects in the Kardar-Parisi-Zhang universality class, namely the Airy sheet, directed landscape and KPZ fixed point. This is joint work with Amol Aggarwal and Milind Hegde. -
Friday, March 1, 202411:10AM, Warren Weaver Hall 1302
A glimpse of universality in critical planar lattice models
Dmitrii Krachun, Princeton UniversitySynopsis:
Many models of statistical mechanics are defined on a lattice, yet
they describe behaviour of objects in our seemingly isotropic world. It is
then natural to ask why, in the small mesh size limit, the directions of the
lattice disappear. Physicists' answer to this question is partially given by
the Universality hypothesis, which roughly speaking states that critical
properties of a physical system do not depend on the lattice or fine
properties of short-range interactions but only depend on the spatial
dimension and the symmetry of the possible spins. Justifying the reasoning
behind the universality hypothesis mathematically seems virtually impossible
and so other ideas are needed for a rigorous derivation of universality even
in the simplest of setups.
In this talk I will explain some ideas behind the recent result which proves
rotational invariance of the FK-percolation model. In doing so, we will see
how rotational invariance is related to universality among a certain
one-dimensional family of planar lattices and how the latter can be proved
using exact integrability of the six-vertex model using Bethe ansatz.Based on joint works with Hugo Duminil-Copin, Karol Kozlowski, Ioan
Manolescu, Mendes Oulamara, and Tatiana Tikhonovskaia. -
Friday, February 23, 202412:45PM, Warren Weaver Hall 1302
Surface sums and Yang-Mills gauge theory
Scott Sheffield, MITSynopsis:
Scott Sheffield (starts at 12:45):
Title: Surface sums and Yang-Mills gauge theory
Abstract: Constructing and understanding the basic properties of Euclidean Yang-Mills theory is a fundamental problem in physics. It is also one of the Clay Institute's famous Millennium Prize problems in mathematics. The basic problem is not hard to understand. You can begin by describing a simple random function from a set of lattice edges to a group of matrices. Then you ask whether you can construct/understand a continuum analog of this object in one way or another. In addition to a truly enormous physics literature, this topic has inspired research within many major areas of mathematics: representation theory, random matrix theory, probability theory, differential geometry, stochastic partial differential equations, low-dimensional topology, graph theory and planar-map combinatorics.Attempts to understand this problem in the 1970's and 1980's helped inspire the study of "random surfaces" including Liouville quantum gravity surfaces. Various relationships between Yang-Mills theory and random surface theory have been obtained over the years, but many of the most basic questions have remained out of reach. I will discuss our own recent work in this direction, as contained in two long recent papers relating "Wilson loop expectations" (the fundamental objects in Yang-Mills gauge theory) to "sums over spanning surfaces."1. Wilson loop expectations as sums over surfaces on the plane (joint with Minjae Park, Joshua Pfeffer, Pu Yu)2. Random surfaces and lattice Yang-Mills (joint with Sky Cao, Minjae Park)The first paper explains how in 2D (where Yang-Mills theory is more tractable) one can interpret continuum Wilson loop expectations purely in terms of flat surfaces. The second explains a general-dimensional interpretation of the Wilson loop expectations in lattice Yang-Mills theory in terms of discrete-and not-necessarily-flat surfaces, a.k.a. embedded planar maps. -
Friday, February 23, 202411:10AM, Warren Weaver Hall 1302
Cutting Liouville quantum gravity by SLE with mismatched parameters
Morris Ang, Columbia UniversitySynopsis:
Beginning with the seminal work of Sheffield, there have been many deep and useful theorems relating Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG) when their parameters are matched, meaning $\kappa \in \{ \gamma^2, 16/\gamma^2\}$. Roughly speaking, the SLE curve cuts the LQG surface into two or more independent LQG surfaces. We extend these theorems to the setting of mismatched parameters: an LQG disk is cut by an SLE curve into two or more LQG surfaces which are conditionally independent given the values along the SLE curve of a certain collection of auxiliary fields. These fields are sampled independently of the LQG and SLE, and have the property that the central charges of the LQG, SLE and auxiliary fields sum to 26. This central charge condition is natural from the perspective of bosonic string theory. Similar statements hold when the SLE curve is replaced by, e.g., an LQG metric ball or a Brownian motion path. These statements are continuum analogs of certain Markov properties of random planar maps decorated by two or more statistical physics models.Based on joint work with Ewain Gwynne.
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Friday, February 16, 202411:10AM, Warren Weaver Hall 1302
Scaling limits for random growth driven by reflecting Brownian motion
Kevin Yang, Harvard UniversitySynopsis:
We discuss long-time asymptotics for a continuum version of origin-excited random walk. It is a growing submanifold in Euclidean space that is pushed outward from within by the boundary trace of a reflecting Brownian motion. We show that the leading-order behavior of the submanifold process is described by a flow-type PDE whose blow-ups correspond to changes in diffeomorphism class of the growth process. We then show that if we simultaneously smooth the submanifold as it grows, fluctuations of an associated height function are described by a regularized KPZ equation with noise modulated by a Dirichlet-to-Neumann operator. If the dimension of the manifold is 2, we show well-posedness of the singular limit of this regularized KPZ-type equation. Based on joint work with Amir Dembo.
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Friday, February 9, 202411AM, Columbia University, Mathematics Hall, 2990 Broadway 207
Viscosity Solutions for HJB Equations on the Process Space: Application to Mean Field Control with Common Noise
Nizar Touzi, NYUSynopsis:
In this paper we investigate a path dependent optimal control problem on the process space with both drift and volatility controls, with possibly degenerate volatility. The dynamic value function is characterized by a fully nonlinear second order path dependent HJB equation on the process space, which is by nature infinite dimensional. In particular, our model covers mean field control problems with common noise as a special case. We introduce a new notion of viscosity solutions and establish both existence and comparison principle, under merely Lipschitz continuity assumptions.
Notes:
Probability and the City seminar
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Friday, February 9, 202410AM, Columbia University, Mathematics Hall, 2990 Broadway 207
Hydrodynamic large deviations of TASEP
Li-Cheng Tsai, University of UtahSynopsis:
Consider the large deviations from the hydrodynamic limit of the Totally Asymmetric Simple Exclusion Process (TASEP), which is related to the entropy production in the inviscid Burgers equation. I will present a result, jointly with Jeremy Quastel, on the full large deviation principle. Our method relies on the explicit formula of Matetski, Quastel, and Remenik (2016) for the transition probabilities of the TASEP.
Notes:
Probability and the City seminar