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* '''When''': Thursdays at 2:30 pm  
* '''When''': Thursdays at 2:30 pm  
* '''Where''': 901 Van Vleck Hall  
* '''Where''': 901 Van Vleck Hall  
* '''Organizers''': Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
* '''Organizers''': Hongchang Ji, Ander Aguirre, Hai-Xiao Wang
* '''To join the probability seminar mailing list:''' email probsem+subscribe@g-groups.wisc.edu.  
* '''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
* '''To subscribe seminar lunch announcements:''' email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu
Line 10: Line 10:
[[Past Seminars]]
[[Past Seminars]]


== Fall 2025 ==


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


== January 23, 2025: ==
== September 4, 2025: No seminar ==
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.
== September 11, 2025: David Renfrew (Binghamton U.) ==


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


== March 13, 2025: Klara Courteaut (Courant) ==
'''Singularities in the spectrum of random block matrices'''
'''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.  
We consider the density of states of structured Hermitian and non-Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up.


Based on joint work with Kurt Johansson and Fredrik Viklund. 
== September 18, 2025: JE Paguyo (McMaster U.) ==
'''Asymptotic behavior of the hierarchical Pitman-Yor and Dirichlet processes'''


== March 20, 2025: Ewain Gwynne (UChicago) ==
The Pitman-Yor process is a discrete random measure specified by a concentration parameter, discount parameter, and base distribution, and is used as a fundamental prior in Bayesian nonparametrics. The hierarchical Pitman-Yor process (HPYP) is a generalization obtained by randomizing the base distribution through a draw from another Pitman-Yor process. It is motivated by the study of groups of clustered data, where the group specific Pitman-Yor processes are linked through an intergroup Pitman-Yor process. Setting both discount parameters to zero recovers the celebrated hierarchical Dirichlet process (HDP), first introduced by Teh et al.
'''Random walk reflected off of infinity''' 
In this talk, we discuss our recent work on the asymptotic behavior of the HPYP and HDP. First, we establish limit theorems associated with the power sum symmetric polynomials for the vector of weights of the HDP as the concentration parameters tend to infinity. These objects are related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl-Hirschman index in economics. Second, we consider a random sample of size $N$ from a population whose type distribution is given by the vector of weights of the HPYP and study the large $N$ asymptotic behavior of the number of clusters in the sample. Our approach relies on a random sample size representation of the number of clusters through the corresponding non-hierarchical process. This talk is based on joint work with Stefano Favaro and Shui Feng.


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.
== September 25, 2025: Chris Janjigian (Purdue U.) ==


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.
== October 2, 2025: Elliot Paquette (McGill U.) ==


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 $c > 95/4$).
== October 9, 2025: No seminar (Midwest Probability Colloquium) ==


Based on joint work with Jinwoo Sung.
== October 16, 2025: Zachary Selk (Florida State U.) ==


== March 27, 2025: SPRING BREAK ==
'''<br />On the Onsager-Machlup Function for the \Phi^4 Measure'''
No seminar 


== April 3, 2025: Jimme He (OSU) ==
The \Phi^4 measure is a measure arising in effective quantum field theory as arguably the simplest example of a nontrivial QFT, modelling the self-interaction of a single scalar quantum field. This measure can be constructed through a procedure known as stochastic quantization. Stochastic quantization seeks to construct a measure on an infinite dimensional space with a given Gibbs-type ``density function" as the invariant measure of a stochastic PDE, in analogy with Langevin dynamics of stochastic ODEs. Both the \Phi^4 measure and its associated stochastic quantization PDE involve nonlinearities of distributions, necessitating renormalization procedures via tools like Wick calculus, regularity structures or paracontrolled calculus. Although the \Phi^4 measure has been constructed in dimensions 1,2 and 3, the question of whether these measures have the desired ``density function" remains open. Although in infinite dimensions, density functions are typically thought to not exist as there is no reference Lebesgue measure, there is a notion of a probability density function that extends to infinite dimensions called the Onsager-Machlup (OM) functional. One pathology of OM theory is that different metrics can lead to different OM functionals, or OM functionals can fail to exist under reasonable metrics. In a joint work with Ioannis Gasteratos (TU Berlin), we study the OM functional for the \Phi^4 measure. In dimension 1, the OM functional is what is desired under naive choices of metrics. In dimension 2, the OM functional is what is desired if we choose a metric analogous to the rough paths metric. In dimension 3, naive approaches don't work and the situation is complicated.
'''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) ==
==October 23, 2025: Alex Dunlap (Duke U.)==
 
'''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.   
==October 30, 2025: Ander Aguirre (UW-Madison)==


== April 17, 2025: ==
'''Edgeworth expansion and random polynomials'''
No seminar 


== April 24, 2025: William Leeb (University of Minnesota, Twin Cities) ==
==November 6, 2025: Sudeshna Bhattacharjee (Indian Institute of Science)==
TBD 


== May 1, 2025: Hai-Xiao Wang (UCSD) ==
== November 13, 2025: Jiaoyang Huang (U. Penn) ==
TBD

Latest revision as of 01:17, 2 September 2025

Back to Probability Group

  • When: Thursdays at 2:30 pm
  • Where: 901 Van Vleck Hall
  • Organizers: Hongchang Ji, Ander Aguirre, Hai-Xiao Wang
  • 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

Fall 2025

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

We usually end for questions at 3:20 PM.

September 4, 2025: No seminar

September 11, 2025: David Renfrew (Binghamton U.)

Singularities in the spectrum of random block matrices

We consider the density of states of structured Hermitian and non-Hermitian random matrices with a variance profile. As the dimension tends to infinity the associated eigenvalue density can develop a singularity at the origin. The severity of this singularity depends on the relative positions of the zero submatrices. We provide a classification of all possible singularities and determine the exponent in the density blow-up.

September 18, 2025: JE Paguyo (McMaster U.)

Asymptotic behavior of the hierarchical Pitman-Yor and Dirichlet processes

The Pitman-Yor process is a discrete random measure specified by a concentration parameter, discount parameter, and base distribution, and is used as a fundamental prior in Bayesian nonparametrics. The hierarchical Pitman-Yor process (HPYP) is a generalization obtained by randomizing the base distribution through a draw from another Pitman-Yor process. It is motivated by the study of groups of clustered data, where the group specific Pitman-Yor processes are linked through an intergroup Pitman-Yor process. Setting both discount parameters to zero recovers the celebrated hierarchical Dirichlet process (HDP), first introduced by Teh et al. In this talk, we discuss our recent work on the asymptotic behavior of the HPYP and HDP. First, we establish limit theorems associated with the power sum symmetric polynomials for the vector of weights of the HDP as the concentration parameters tend to infinity. These objects are related to the homozygosity in population genetics, the Simpson diversity index in ecology, and the Herfindahl-Hirschman index in economics. Second, we consider a random sample of size $N$ from a population whose type distribution is given by the vector of weights of the HPYP and study the large $N$ asymptotic behavior of the number of clusters in the sample. Our approach relies on a random sample size representation of the number of clusters through the corresponding non-hierarchical process. This talk is based on joint work with Stefano Favaro and Shui Feng.

September 25, 2025: Chris Janjigian (Purdue U.)

October 2, 2025: Elliot Paquette (McGill U.)

October 9, 2025: No seminar (Midwest Probability Colloquium)

October 16, 2025: Zachary Selk (Florida State U.)


On the Onsager-Machlup Function for the \Phi^4 Measure

The \Phi^4 measure is a measure arising in effective quantum field theory as arguably the simplest example of a nontrivial QFT, modelling the self-interaction of a single scalar quantum field. This measure can be constructed through a procedure known as stochastic quantization. Stochastic quantization seeks to construct a measure on an infinite dimensional space with a given Gibbs-type ``density function" as the invariant measure of a stochastic PDE, in analogy with Langevin dynamics of stochastic ODEs. Both the \Phi^4 measure and its associated stochastic quantization PDE involve nonlinearities of distributions, necessitating renormalization procedures via tools like Wick calculus, regularity structures or paracontrolled calculus. Although the \Phi^4 measure has been constructed in dimensions 1,2 and 3, the question of whether these measures have the desired ``density function" remains open. Although in infinite dimensions, density functions are typically thought to not exist as there is no reference Lebesgue measure, there is a notion of a probability density function that extends to infinite dimensions called the Onsager-Machlup (OM) functional. One pathology of OM theory is that different metrics can lead to different OM functionals, or OM functionals can fail to exist under reasonable metrics. In a joint work with Ioannis Gasteratos (TU Berlin), we study the OM functional for the \Phi^4 measure. In dimension 1, the OM functional is what is desired under naive choices of metrics. In dimension 2, the OM functional is what is desired if we choose a metric analogous to the rough paths metric. In dimension 3, naive approaches don't work and the situation is complicated.


October 23, 2025: Alex Dunlap (Duke U.)

October 30, 2025: Ander Aguirre (UW-Madison)

Edgeworth expansion and random polynomials

November 6, 2025: Sudeshna Bhattacharjee (Indian Institute of Science)

November 13, 2025: Jiaoyang Huang (U. Penn)

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