Probability Seminar

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Past Seminars

Fall 2023

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

September 14, 2023: Matthew Junge (CUNY)

The frog model on trees

The frog model describes random activation and spread. Think combustion or an epidemic. I have studied these dynamics on d-ary trees for ten years. I will discuss our progress and what remains to be done.

September 21, 2023: Yier Lin (U. Chicago)

Large Deviations of the KPZ Equation and Most Probable Shapes

The KPZ equation is a stochastic PDE that plays a central role in a class of random growth phenomena. In this talk, we will explore the Freidlin-Wentzell LDP for the KPZ equation through the lens of the variational principle. Additionally, we will explain how to extract various limits of the most probable shape of the KPZ equation using the variational formula. We will also discuss an alternative approach for studying these quantities using the method of moments. This talk is based in part on joint works with Pierre Yves Gaudreau Lamarre and Li-Cheng Tsai.

September 28, 2023: Tommaso Rosati (U. Warwick)

The Allen-Cahn equation with weakly critical initial datum

We study the 2D Allen-Cahn with white noise initial datum. In a weak coupling regime, where the nonlinearity is damped in relation to the smoothing of the initial condition, we prove Gaussian fluctuations. The effective variance that appears can be described as the solution to an ODE. Our proof builds on a Wild expansion of the solution, which is controlled through precise combinatorial estimates. Joint work with Simon Gabriel and Nikolaos Zygouras.

October 5, 2023:

Abstract, title: TBA

October 19, 2023: Paul Duncan (Hebrew University of Jerusalem)

Deconfinement in Ising Lattice Gauge Theory

A lattice gauge theory is a random assignment of spins to edges of a lattice that offers a more tractable model in which to study path integrals that appear in particle physics. We demonstrate the existence of a phase transition corresponding to deconfinement in a simplified model called Ising lattice gauge theory on the cubical lattice Z^3. Our methods involve studying the topology of a random 2-dimensional cubical complex on Z^3 called random-cluster plaquette percolation, which in turn can be reduced to the study of a random dual graph. No prior background in topology or physics will be assumed. This is based on joint work with Benjamin Schweinhart.

October 26, 2023: Yuchen Liao (UW - Madison)

Large deviations for the deformed Polynuclear growth

The polynuclear growth model (PNG) is a prototypical example of random interface growth among the Kardar-Parisi-Zhang universality class. In this talk I will discuss a q-deformation of the PNG model recently introduced by Aggarwal-Borodin-Wheeler. We are mainly interested in the large time large deviations of the one-point distribution under narrow-wedge (droplet) initial data, i.e., the rare events that the height function at time t being much larger (upper tail) or much smaller (lower tail) than its expected value. Large deviation principles with speed t and t^2 are established for the upper and lower tails, respectively. The upper tail rate function is computed explicitly and is independent of q. The lower tail rate function is described through a variational problem and shows nontrivial q-dependence.  Based on joint  work with Matteo Mucciconi and Sayan Das.

November 2, 2023: Cheng Ouyang (U. Illinois Chicago)

Colored noise and parabolic Anderson model on Torus

We construct an intrinsic family of Gaussian noises on compact Riemannian manifolds which we call the colored noise on manifolds. It consists of noises with a wide range of singularities. Using this family of noises, we study the parabolic Anderson model on compact manifolds. To begin with, we started our investigation on a flat torus and established existence and uniqueness of the solution, as well as some sharp bounds on the second moment of the solution. In particular, our methodology does not necessarily rely on Fourier analysis and can be applied to study the PAM on more general manifolds.

November 9, 2023: Scott Smith (Chinese Academy of Sciences)

A stochastic analysis viewpoint on the master loop equation for lattice Yang-Mills

I will discuss the master loop equation for lattice Yang-Mills, introduced in the physics literature by Makeenko/Migdal (1979).  A more precise formulation and proof was given by Chatterjee (2019) for SO(N) and later by Jafarov for SU(N).  I will explain how the loop equation arises naturally from the Langevin dynamic for the lattice Yang-Mills measure.  Based on joint work with Hao Shen and Rongchan Zhu.

November 16, 2023: Matthew Nicoletti (MIT)

Colored Interacting Particle Systems on the Ring: Stationary Measures from Yang--Baxter Equation

Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials).

    In this work, we present a unified approach to constructing stationary measures for several colored particle systems on the ring and the line, including (1)~the Asymmetric Simple Exclusion Process (mASEP); (2)~the $q$-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the $q$-Boson particle system; (3)~the $q$-deformed Pushing Totally Asymmetric Simple Exclusion Process ($q$-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang--Baxter equation. We express the stationary measures as partition functions of new ``queue vertex models'' on the cylinder. The stationarity property is a direct consequence of the Yang--Baxter equation. This is joint work with A. Aggarwal and L. Petrov.

November 23, 2023: No Seminar

No seminar. Thanksgiving.

November 30, 2023: Youngtak Sohn (MIT)

Geometry of random constraint satisfaction problems

The framework of constraint satisfaction problem (CSP) captures many fundamental problems in combinatorics and computer science, such as finding a proper coloring of a graph or solving the boolean satisfiability problems. Solving a CSP can often be NP-hard in the worst-case scenario. To study the typical cases of CSPs, statistical physicists have proposed a detailed picture of the solution space for random CSPs based on non-rigorous methods from spin glass theory. In this talk, I will first survey the conjectured rich phase diagrams of random CSPs in the one-step replica symmetry breaking (1RSB) universality class. Then, I will describe the recent progress in understanding the global and local geometry of solutions, particularly in random regular NAE-SAT problem.

This talk is based on joint works with Danny Nam and Allan Sly.

December 7, 2023: Minjae Park (U. Chicago)

Abstract, title: TBA