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[[Probability | Back to Probability Group]]
[[Probability | Back to Probability Group]]
* '''When''': Thursdays at 2:30 pm
* '''Where''': 901 Van Vleck Hall
* '''Organizers''': Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
* '''To join the probability seminar mailing list:''' email probsem+subscribe@g-groups.wisc.edu.
* '''To subscribe seminar lunch announcements:''' email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu


[[Past Seminars]]
[[Past Seminars]]


= Spring 2023 =


<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>  
 
= Spring 2025 =
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>


We usually end for questions at 3:20 PM.
We usually end for questions at 3:20 PM.


[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]
== January 23, 2025: ==
No seminar 


If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].
== January 30, 2025: Promit Ghosal (UChicago) ==
'''Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond''' 


Stein Variational Gradient Descent (SVGD) has emerged as a powerful interacting particle-based algorithm for nonparametric sampling, yet its theoretical properties remain challenging to unravel. This talk delves into two complementary perspectives about SVGD. First, we explore Gaussian-SVGD, a framework that projects SVGD onto the family of Gaussian distributions via a bilinear kernel. We establish rigorous convergence results for both mean-field dynamics and finite-particle systems, demonstrating linear convergence to equilibrium in strongly log-concave settings and unifying recent algorithms for Gaussian variational inference (GVI) under a single framework. Second, we analyze the finite-particle convergence rates of SVGD in Kernelized Stein Discrepancy (KSD) and Wasserstein-2 metrics. Leveraging a novel decomposition of the relative entropy time derivative, we achieve near-optimal rates with polynomial dimensional dependence and extend these results to bilinear-enhanced kernels.


== January 26, 2023, in person: [https://sites.google.com/wisc.edu/evan-sorensen?pli=1 Evan Sorensen] (UW-Madison)   ==
== February 6, 2025: Subhabrata Sen (Harvard) ==
'''The stationary horizon as a universal object for KPZ models'''
'''Community detection on multi-view networks'''
 
The last 5-10 years has seen remarkable progress in constructing the central objects of the KPZ universality class, namely the KPZ fixed point and directed landscape. In this talk, I will discuss a third central object known as the stationary horizon (SH). The SH is a coupling of Brownian motions with drifts, indexed by the real line, and it describes the unique coupled invariant measures for the directed landscape. I will talk about how the SH appears as the scaling limit of several models, including Busemann processes in last-passage percolation and the TASEP speed process. I will also discuss how the SH helps to describe the collection of infinite geodesics in all directions for the directed landscape. Based on joint work with Timo Seppäläinen and Ofer Busani.
The community detection problem seeks to recover a latent clustering of vertices from an observed random graph. This problem has attracted significant attention across probability, statistics and computer science, and the fundamental thresholds for community recovery have been characterized in the last decade. Modern applications typically collect more fine-grained information on the units under study. For example, one might measure relations of multiple types among the units, or observe an evolving network over time. In this talk, we will discuss the community detection problem on such ‘multi-view’ networks. We will present some new results on the fundamental thresholds for community detection in these models. Finally, we will introduce algorithms for community detection based on Approximate Message Passing.
 
This is based on joint work with Xiaodong Yang and Buyu Lin (Harvard University).


== February 2, 2023, in person: [https://mathjinsukim.com/ Jinsu Kim] (POSTECH)   ==
== February 13, 2025: Hanbaek Lyu (UW-Madison) ==
'''Fast and slow mixing of continuous-time Markov chains with polynomial rates'''
'''Large random matrices with given margins'''   
   
Continuous-time Markov chains on infinite positive integer grids with polynomial rates are often used in modeling queuing systems, molecular counts of small-size biological systems, etc. In this talk, we will discuss continuous-time Markov chains that admit either fast or slow mixing behaviors. For a positive recurrent continuous-time Markov chain, the convergence rate to its stationary distribution is typically investigated with the Lyapunov function method and canonical path method. Recently, we discovered examples that do not lend themselves easily to analysis via those two methods but are shown to have either fast mixing or slow mixing with our new technique. The main ideas of the new methodologies are presented in this talk along with their applications to stochastic biochemical reaction network theory.


== February 9, 2023, in person: [https://www.math.tamu.edu/~jkuan/ Jeffrey Kuan] (Texas A&M)   ==
We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization,  Schr\"{o}dinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. We show that such a margin-constrained random matrix is sharply concentrated around a certain deterministic matrix, which we call the ''typical table''. Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two potential functions, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in $L^{1}$ to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$.  We also propose a generalized Sinkhorn algorithm for computing typical tables and establish its linear convergence. We derive several new results for random contingency tables from our general framework. 
'''Shift invariance for the multi-species q-TAZRP on the infinite line'''


We prove a shift--invariance for the multi-species q-TAZRP (totally asymmetric zero range process) on the infinite line. Similar-looking results had appeared in works by [Borodin-Gorin-Wheeler] and [Galashin], using integrability, but are on the quadrant. The proof in this talk relies instead on a combinatorial approach, in which the state space is generalized to a poset, and the totally asymmetric process is generalized to a monotone process on a poset. The continuous-time process is decomposed into its discrete embedded Markov chain and its exponential holding times, and the shift-invariance is proved using explicit contour integral formulas. Open problems about multi-species ASEP will be discussed as well.
Based on a joint work with Sumit Mukherjee (Columbia)


== February 16, 2023, in person: [http://math.columbia.edu/~milind/ Milind Hegde] (Columbia)   ==
== February 20, 2025: Mustafa Alper Gunes (Princeton) ==
'''Understanding the upper tail behaviour of the KPZ equation via the tangent method'''
'''Characteristic Polynomials of Random Matrices, Exchangeable Arrays & Painlevé Equations'''


The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, related to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data.  
Joint moments of characteristic polynomials of unitary random matrices and their derivatives have gained attention over the last 25 years, partly due to their conjectured relation to the Riemann zeta function. In this talk, we will consider the asymptotics of these moments in the most general setting allowing for derivatives of arbitrary order, generalising previous work that considered only the first derivative. Along the way, we will examine how exchangeable arrays and integrable systems play a crucial role in understanding the statistics of a class of infinite Hermitian random matrices. Based on joint work with Assiotis, Keating and Wei.


== February 23, 2023, in person: [https://sites.math.rutgers.edu/~sc2518/ Swee Hong Chan] (Rutgers)   ==
== February 27, 2025: Souvik Dhara (Purdue) ==
'''Log-concavity and cross product inequalities in order theory'''
'''Propagation of Shocks on Networks: Can Local Information Predict Survival?'''  


Given a finite poset that is not completely ordered, is it always possible find two elements x and y, such that the probability that x is less than y in the random linear extension of the poset, is bounded away from 0 and 1? Kahn-Saks gave an affirmative answer and showed that this probability falls between 3/11 (0.273) and 8/11 (0.727). The currently best known bound is 0.276 and 0.724 by Brightwell-Felsner-Trotter, and it is believed that the optimal bound should be 1/3 and 2/3, also known as the 1/3-2/3 Conjecture. Most notably, log-concave and cross product inequalities played the central role in deriving both bounds. In this talk we will discuss various generalizations of these results together with related open problems. This talk is joint work with Igor Pak and Greta Panova, and is intended for the general audience.
Abstract: Complex systems are often fragile, where minor disruptions can cascade into dramatic collapses. Epidemics serve as a prime example of this phenomenon, while the 2008 financial crisis highlights how a domino effect, originating from the small subprime mortgage sector, can trigger global repercussions. The mathematical theory underlying these phenomena is both elegant and foundational, profoundly shaping the field of Network Science since its inception. In this talk, I will present a unifying mathematical model for network fragility and cascading dynamics, and explore its deep connections to the theory of local-weak convergence, pioneered by Benjamini-Schramm and Aldous-Steele.


== March 2, 2023, in person: Max Hill (UW-Madison)   ==
== March 6, 2025: Alexander Meehan (UW-Madison, Department of Philosophy) ==
'''On the Effect of Intralocus Recombination on Triplet-Based Species Tree Estimation'''
'''What conditional probability could (probably) be'''  


My talk will introduce some key topics in mathematical phylogenetics and is intended to be accessible for those not familiar with the field. I will discuss joint work with Sebastien Roch on the subject of species tree estimation from multiple loci subject to intralocus recombination. The focus is on R*, a summary coalescent-based method using rooted triplets. I will present a result showing how intralocus recombination can give rise to an "inconsistency zone," in which correct inference using R* is not assured even in the limit of infinite amount of data.
According to orthodox probability theory, when B has probability zero, the conditional probability of A given B can depend on the partition or sub-sigma-field that B is relativized to. This relativization to sub-sigma-fields, a hallmark of Kolmogorov's theory of conditional expectation, is traditionally seen as appropriate in a treatment of conditioning with continuous variables, and it is what allows the theory to preserve Total Disintegrability, a generalization of the Law of Total Probability to uncountable partitions. In this talk, I will argue that although the relativization of conditional probability to sub-sigma-fields has advantages, it also has an underrecognized cost: it leads to puzzles for the treatment of ''iterated conditioning''. I will discuss these puzzles and some possible implications for the foundations of conditional probability.


== March 9, 2023, in person: [https://math.uchicago.edu/~xuanw/ Xuan Wu] (U. Chicago)    ==
This talk is based on joint work with Snow Zhang (UC Berkeley).  
'''From the KPZ equation to the directed landscape'''


This talk presents the convergence of the KPZ equation to the directed landscape, which is the central object in the KPZ universality class. This convergence result is the first to the directed landscape among the positive temperature models.
== March 13, 2025: Klara Courteaut (Courant) ==
'''The Coulomb gas on a Jordan arc''' 


== March 23, 2023, in person: Jiaming Xu (UW-Madison)    ==
We study a Coulomb gas on a sufficiently smooth simple arc in the complex plane, at arbitrary positive temperature. We show that as the number of particles tends to infinity, the partition function converges to a quantity involving the partition function of the log-gas on [−1,1] and the Fredholm determinant of the arc-Grunsky operator. Alternatively, we can express this quantity in terms of the Loewner energy of a specific Jordan curve associated with the arc. We also obtain an asymptotic formula for the Laplace transform of linear statistics for sufficiently regular test functions. This shows that the centered empirical measure converges to a Gaussian field with explicit asymptotic mean and asymptotic variance given by the Dirichlet energy of the test function. 


'''Rectangular Matrix addition in low and high temperatures'''
Based on joint work with Kurt Johansson and Fredrik Viklund. 


We study the addition of two <math>{\scriptsize M \times N}</math> rectangular random matrices with certain
== March 20, 2025: Ewain Gwynne (UChicago) ==
invariant distributions in two limit regimes, where the parameter <math>{\scriptsize \beta}</math> (inverse temperature) goes to infinity and zero. In low temperature regime the random singular values of the sum concentrate at deterministic points, while in high temperature regime we obtain a Law of Large Numbers of the empirical measures. Our proof uses the so-called type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in high temperature limit, and
'''Random walk reflected off of infinity''' 
degenerate to the classical or free cumulants in special cases.


== March 30, 2023, in person: [http://www.math.toronto.edu/balint/ Bálint Virág] (Toronto)    ==
Let $\mathcal G$ be an infinite graph --- not necessarily one-ended --- on which the simple random walk is transient. We define a variant of the continuous-time random walk on $\mathcal G$ which reaches $\infty$ in finite time and ``reflects off of $\infty$<nowiki>''</nowiki> infinitely many times.
'''The planar stochastic heat equation and the directed landscape'''


The planar stochastic heat equation describes heat flow or random polymers on an inhomogeneous surface. It is a finite-temperature version of planar first passage percolation such as the Eden growth model. It is the first model with plane symmetries for which we can show convergence to the directed landscape. The methods use a Skorokhod integral representation and Gaussian multiplicative chaos on path space.  
We show that the Aldous-Broder algorithm for the random walk reflected off of $\infty$ gives the free uniform spanning forest (FUSF) on $\mathcal G$. Furthermore, Wilson's algorithm for the random walk reflected off of $\infty$ gives the FUSF on $\mathcal G$ on the event that the FUSF is connected, but not in general.


Joint work with Jeremy Quastel and Alejandro Ramirez.
We also apply the theory of random walk reflected off of $\infty$ to study random planar maps in the universality class of supercritical Liouville quantum gravity (LQG), equivalently LQG with central charge $c \in (1,25)$. Such random planar maps are infinite, with uncountably many ends. We define a version of the Tutte embedding for such maps under which they conjecturally converge to LQG. We also conjecture that the free uniform spanning forest on these maps is connected when $c > 16$ (but not when $c < 16$); and that there is an infinite open cluster for critical percolation on these maps when $c < 95/4$ (but not when $c > 95/4$).  


== April 6, 2023, in person: [https://shankarbhamidi.web.unc.edu/ Shankar Bhamidi] (UNC-Chapel Hill)    ==
Based on joint work with Jinwoo Sung.


'''Disorder models for random graphs, Erdos’s leader problem, and power of limited choice models for network evolution'''
== March 27, 2025: SPRING BREAK ==
No seminar  
First passage percolation, and more generally the study of diffusion of material through disordered systems is a fundamental area in probabilistic combinatorics with a vast body of work especially in the context of spatial systems.
The goal of this talk is to survey a slightly different setting for such questions namely the more “mean-field” setting of random graph models. We will describe the state of the art of this field, with the final goal of describing one of the main conjectures in this area namely the conjectured scaling limit of the minimal spanning tree and its dependence on the degree exponent of the corresponding network model. We will describe recent progress in this area, its connection to questions in dynamic network models, in particular Erdos’s leader problem for the identity of the maximal component for critical random graphs, and the intuition for understanding the evolution of maximal components through the critical scaling window from a different area of probabilistic combinatorics, namely the study of limited choice models for network evolution.  


== April 13, 2023, in person: [http://www.bricehuang.com/index.html Brice Huang] (MIT)   ==
== April 3, 2025: Jimme He (OSU) ==
'''Algorithmic Threshold for Multi-Species Spherical Spin Glasses'''
'''Random growth models with half space geometry'''


This talk focuses on optimizing the random and non-convex Hamiltonians of spherical spin glasses with multiple species. Our main result identifies the best possible value ALG achievable by class of Lipschitz algorithms and gives a matching algorithm in this class based on approximate message passing. The threshold ALG is given by a certain variational problem, which surprisingly may possess multiple optimizers.
Abstract: Random growth models in 1+1 dimension capture the behavior of interfaces evolving in the presence of noise. These models are expected to exhibit universal behavior, but we are still far from proving such results even in relatively simple models. A key development which has led to recent progress is the discovery of exact formulas for certain models with rich algebraic structure, leading to asymptotic results. I will discuss work on the asymmetric simple exclusion process with one open boundary, as well as applications to rates of convergence for a Markov chain.


Our hardness result is proved using the Branching OGP introduced in our previous work [H-Sellke 21] to identify ALG for single-species spin glasses. This and all other OGPs for spin glasses have been proved using Guerra's interpolation method. We introduce a new method to prove the Branching OGP which is both simpler and more robust. It works even for models in which the true maximum value of the objective function remains unknown.
== April 10, 2025: Evan Sorensen (Columbia) ==
 
'''Viscous shock fluctuations in KPZ''' 


Based on joint work with Mark Sellke.
I will discuss a recent preprint with Alex Dunlap, where we study ``V-shaped" solutions to the KPZ equation. These are solutions having asymptotic slopes \theta > 0 and -\theta at plus and minus infinity, respectively. We show that there are no V-shaped invariant measures for the KPZ equation, which, combined with recent work of Janjigian, Rassoul-Agha, and Seppalainen, completes the classification of the extremal invariant measures for the KPZ equation. To accomplish this, we study the fluctuations of viscous shocks in the KPZ equation under some special choices of initial data. While V-shaped invariant measures in a fixed frame of reference do not exist, we give an explicit description of a family of V-shaped invariant measures from the perspective of a shock.   


== April 20, 2023, in person: [http://www.math.columbia.edu/~remy/ Guillaume Remy] (IAS)    ==
== April 17, 2025: ==
No seminar 


== April 27, 2023, in person: [http://www.math.tau.ac.il/~peledron/ Ron Peled] (Tel Aviv/IAS)   ==
== April 24, 2025: William Leeb (University of Minnesota, Twin Cities) ==
TBD 


== May 4, 2023, in person: [https://www.asc.ohio-state.edu/sivakoff.2// David Sivakoff] (Ohio State)   ==
== May 1, 2025: Hai-Xiao Wang (UCSD) ==
TBD

Latest revision as of 17:32, 1 April 2025

Back to Probability Group

  • When: Thursdays at 2:30 pm
  • Where: 901 Van Vleck Hall
  • Organizers: Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
  • To join the probability seminar mailing list: email probsem+subscribe@g-groups.wisc.edu.
  • To subscribe seminar lunch announcements: email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu

Past Seminars


Spring 2025

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

We usually end for questions at 3:20 PM.

January 23, 2025:

No seminar

January 30, 2025: Promit Ghosal (UChicago)

Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond

Stein Variational Gradient Descent (SVGD) has emerged as a powerful interacting particle-based algorithm for nonparametric sampling, yet its theoretical properties remain challenging to unravel. This talk delves into two complementary perspectives about SVGD. First, we explore Gaussian-SVGD, a framework that projects SVGD onto the family of Gaussian distributions via a bilinear kernel. We establish rigorous convergence results for both mean-field dynamics and finite-particle systems, demonstrating linear convergence to equilibrium in strongly log-concave settings and unifying recent algorithms for Gaussian variational inference (GVI) under a single framework. Second, we analyze the finite-particle convergence rates of SVGD in Kernelized Stein Discrepancy (KSD) and Wasserstein-2 metrics. Leveraging a novel decomposition of the relative entropy time derivative, we achieve near-optimal rates with polynomial dimensional dependence and extend these results to bilinear-enhanced kernels.

February 6, 2025: Subhabrata Sen (Harvard)

Community detection on multi-view networks

The community detection problem seeks to recover a latent clustering of vertices from an observed random graph. This problem has attracted significant attention across probability, statistics and computer science, and the fundamental thresholds for community recovery have been characterized in the last decade. Modern applications typically collect more fine-grained information on the units under study. For example, one might measure relations of multiple types among the units, or observe an evolving network over time. In this talk, we will discuss the community detection problem on such ‘multi-view’ networks. We will present some new results on the fundamental thresholds for community detection in these models. Finally, we will introduce algorithms for community detection based on Approximate Message Passing.

This is based on joint work with Xiaodong Yang and Buyu Lin (Harvard University).

February 13, 2025: Hanbaek Lyu (UW-Madison)

Large random matrices with given margins

We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margin). This problem has rich connections to relative entropy minimization,  Schr\"{o}dinger bridge, the enumeration of contingency tables, and random graphs with given degree sequences. We show that such a margin-constrained random matrix is sharply concentrated around a certain deterministic matrix, which we call the typical table. Typical tables have dual characterizations: (1) the expectation of the random matrix ensemble with minimum relative entropy from the base model constrained to have the expected target margin, and (2) the expectation of the maximum likelihood model obtained by rank-one exponential tilting of the base model. The structure of the typical table is dictated by two potential functions, which give the maximum likelihood estimates of the tilting parameters. Based on these results, for a sequence of "tame" margins that converges in $L^{1}$ to a limiting continuum margin as the size of the matrix diverges, we show that the sequence of margin-constrained random matrices converges in cut norm to a limiting kernel, which is the $L^{2}$-limit of the corresponding rescaled typical tables. The rate of convergence is controlled by how fast the margins converge in $L^{1}$.  We also propose a generalized Sinkhorn algorithm for computing typical tables and establish its linear convergence. We derive several new results for random contingency tables from our general framework.

Based on a joint work with Sumit Mukherjee (Columbia)

February 20, 2025: Mustafa Alper Gunes (Princeton)

Characteristic Polynomials of Random Matrices, Exchangeable Arrays & Painlevé Equations

Joint moments of characteristic polynomials of unitary random matrices and their derivatives have gained attention over the last 25 years, partly due to their conjectured relation to the Riemann zeta function. In this talk, we will consider the asymptotics of these moments in the most general setting allowing for derivatives of arbitrary order, generalising previous work that considered only the first derivative. Along the way, we will examine how exchangeable arrays and integrable systems play a crucial role in understanding the statistics of a class of infinite Hermitian random matrices. Based on joint work with Assiotis, Keating and Wei.

February 27, 2025: Souvik Dhara (Purdue)

Propagation of Shocks on Networks: Can Local Information Predict Survival?

Abstract: Complex systems are often fragile, where minor disruptions can cascade into dramatic collapses. Epidemics serve as a prime example of this phenomenon, while the 2008 financial crisis highlights how a domino effect, originating from the small subprime mortgage sector, can trigger global repercussions. The mathematical theory underlying these phenomena is both elegant and foundational, profoundly shaping the field of Network Science since its inception. In this talk, I will present a unifying mathematical model for network fragility and cascading dynamics, and explore its deep connections to the theory of local-weak convergence, pioneered by Benjamini-Schramm and Aldous-Steele.

March 6, 2025: Alexander Meehan (UW-Madison, Department of Philosophy)

What conditional probability could (probably) be

According to orthodox probability theory, when B has probability zero, the conditional probability of A given B can depend on the partition or sub-sigma-field that B is relativized to. This relativization to sub-sigma-fields, a hallmark of Kolmogorov's theory of conditional expectation, is traditionally seen as appropriate in a treatment of conditioning with continuous variables, and it is what allows the theory to preserve Total Disintegrability, a generalization of the Law of Total Probability to uncountable partitions. In this talk, I will argue that although the relativization of conditional probability to sub-sigma-fields has advantages, it also has an underrecognized cost: it leads to puzzles for the treatment of iterated conditioning. I will discuss these puzzles and some possible implications for the foundations of conditional probability.

This talk is based on joint work with Snow Zhang (UC Berkeley).

March 13, 2025: Klara Courteaut (Courant)

The Coulomb gas on a Jordan arc

We study a Coulomb gas on a sufficiently smooth simple arc in the complex plane, at arbitrary positive temperature. We show that as the number of particles tends to infinity, the partition function converges to a quantity involving the partition function of the log-gas on [−1,1] and the Fredholm determinant of the arc-Grunsky operator. Alternatively, we can express this quantity in terms of the Loewner energy of a specific Jordan curve associated with the arc. We also obtain an asymptotic formula for the Laplace transform of linear statistics for sufficiently regular test functions. This shows that the centered empirical measure converges to a Gaussian field with explicit asymptotic mean and asymptotic variance given by the Dirichlet energy of the test function.

Based on joint work with Kurt Johansson and Fredrik Viklund.

March 20, 2025: Ewain Gwynne (UChicago)

Random walk reflected off of infinity

Let $\mathcal G$ be an infinite graph --- not necessarily one-ended --- on which the simple random walk is transient. We define a variant of the continuous-time random walk on $\mathcal G$ which reaches $\infty$ in finite time and ``reflects off of $\infty$'' infinitely many times.

We show that the Aldous-Broder algorithm for the random walk reflected off of $\infty$ gives the free uniform spanning forest (FUSF) on $\mathcal G$. Furthermore, Wilson's algorithm for the random walk reflected off of $\infty$ gives the FUSF on $\mathcal G$ on the event that the FUSF is connected, but not in general.

We also apply the theory of random walk reflected off of $\infty$ to study random planar maps in the universality class of supercritical Liouville quantum gravity (LQG), equivalently LQG with central charge $c \in (1,25)$. Such random planar maps are infinite, with uncountably many ends. We define a version of the Tutte embedding for such maps under which they conjecturally converge to LQG. We also conjecture that the free uniform spanning forest on these maps is connected when $c > 16$ (but not when $c < 16$); and that there is an infinite open cluster for critical percolation on these maps when $c < 95/4$ (but not when $c > 95/4$).

Based on joint work with Jinwoo Sung.

March 27, 2025: SPRING BREAK

No seminar

April 3, 2025: Jimme He (OSU)

Random growth models with half space geometry

Abstract: Random growth models in 1+1 dimension capture the behavior of interfaces evolving in the presence of noise. These models are expected to exhibit universal behavior, but we are still far from proving such results even in relatively simple models. A key development which has led to recent progress is the discovery of exact formulas for certain models with rich algebraic structure, leading to asymptotic results. I will discuss work on the asymmetric simple exclusion process with one open boundary, as well as applications to rates of convergence for a Markov chain.

April 10, 2025: Evan Sorensen (Columbia)

Viscous shock fluctuations in KPZ

I will discuss a recent preprint with Alex Dunlap, where we study ``V-shaped" solutions to the KPZ equation. These are solutions having asymptotic slopes \theta > 0 and -\theta at plus and minus infinity, respectively. We show that there are no V-shaped invariant measures for the KPZ equation, which, combined with recent work of Janjigian, Rassoul-Agha, and Seppalainen, completes the classification of the extremal invariant measures for the KPZ equation. To accomplish this, we study the fluctuations of viscous shocks in the KPZ equation under some special choices of initial data. While V-shaped invariant measures in a fixed frame of reference do not exist, we give an explicit description of a family of V-shaped invariant measures from the perspective of a shock.  

April 17, 2025:

No seminar

April 24, 2025: William Leeb (University of Minnesota, Twin Cities)

TBD

May 1, 2025: Hai-Xiao Wang (UCSD)

TBD

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