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[[Probability | Back to Probability Group]]


= Spring 2021 =
* '''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


<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.
[[Past Seminars]]
<b>We  usually end for questions at 3:20 PM.</b>


<b> IMPORTANT: </b> In Spring 2021 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]


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 28, 2021, no seminar  ==


== February 4, 2021, [https://cims.nyu.edu/~hbchen/ Hong-Bin Chen] (Courant Institute, NYU) ==
= Spring 2025 =
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>


'''Dynamic polymers: invariant measures and ordering by noise'''
We usually end for questions at 3:20 PM.


We develop a dynamical approach to infinite volume polymer measures (IVPM) in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow, and establish ordering by noise. We prove that, for a fixed asymptotic slope, the polymer dynamics has a unique invariant distribution given by a unique IVPM. Moreover, One Force-One Solution principle holds.
== January 23, 2025: ==
No seminar 


== February 11, 2021, [https://mathematics.stanford.edu/people/kevin-yang Kevin Yang] (Stanford) ==
== January 30, 2025: Promit Ghosal (UChicago) ==
'''Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond''' 


'''Non-stationary fluctuations for some non-integrable models'''
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.


We will discuss recent progress on weak KPZ universality and non-integrable particle systems, including long-range models and slow bond models. The approach is based on a preliminary step in a non-stationary (first-order) Boltzmann-Gibbs principle. We will also discuss the full non-stationary Boltzmann-Gibbs principle itself and pieces of its proof.
== February 6, 2025: Subhabrata Sen (Harvard) ==
'''Community detection on multi-view networks''' 


== February 18, 2021, [https://ilyachevyrev.wordpress.com Ilya Chevyrev] (Edinburgh) ==
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.


'''Signature moments to characterize laws of stochastic processes'''
This is based on joint work with Xiaodong Yang and Buyu Lin (Harvard University). 


The normalized sequence of moments characterizes the law of any finite-dimensional random variable. In this talk, I will describe an extension of this result to path-valued random variables, i.e. stochastic processes, by using the normalized sequence of signature moments. I will show how these moments define a metric for laws of stochastic processes. This metric can be efficiently estimated from finite samples, even if the stochastic processes themselves evolve in high-dimensional state spaces. As an application, I will describe a non-parametric two-sample hypothesis test for laws of stochastic processes.
== February 13, 2025: Hanbaek Lyu (UW-Madison) ==
'''Large random matrices with given margins''' 


== February 25, 2021, [https://math.mit.edu/directory/profile.php?pid=2121 Roger Van Peski] (MIT) ==
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.   


'''Random matrices, random groups, singular values, and symmetric functions'''
Based on a joint work with Sumit Mukherjee (Columbia) 


Since the 1989 work of Friedman-Washington, the cokernels of random p-adic matrices drawn from various distributions have provided models for random finite abelian p-groups arising in number theory and combinatorics, the most famous being the class groups of quadratic imaginary number fields. Since any finite abelian p-group is isomorphic to a direct sum of cyclic groups $\bigoplus_i \mathbb{Z}/p^{\lambda_i}\mathbb{Z}$, it is equivalent to study the random integer partition $\lambda = (\lambda_1, \lambda_2,\ldots)$, which is analogous to the singular values of a complex random matrix. We show that the behavior of such partitions under taking products and corners of random p-adic matrices is governed by the Hall-Littlewood polynomials, recovering and explaining some previous results relating p-adic matrix cokernels to these polynomials. We use these exact results to study the joint asymptotic behavior of the cokernels of products of many random p-adic matrices $A_\tau \cdots A_1$, with $\tau$ acting as a discrete time parameter. We show that the parts $\lambda_i$ of the corresponding partition have a simple description via an interacting particle system, and their fluctuations converge under rescaling to independent Brownian motions. At both the exact and asymptotic level we explain connections between our results and existing results on singular values of complex random matrices: both are in fact degenerations of the same operations on random partitions coming from Macdonald polynomials.
== February 20, 2025: Mustafa Alper Gunes (Princeton) ==
'''Characteristic Polynomials of Random Matrices, Exchangeable Arrays & Painlevé Equations''' 


== March 4, 2021, [http://www.statslab.cam.ac.uk/~rb812/ Roland Bauerschmidt] (Cambridge) ==
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.


'''The Coleman correspondence at the free fermion point'''
== February 27, 2025: Souvik Dhara (Purdue) ==
'''Propagation of Shocks on Networks: Can Local Information Predict Survival?'''  


Two-dimensional statistical and quantum field theories are special in many ways. One striking instance of this is the equivalence of certain bosonic and fermionic fields, known as bosonization.
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.
I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions.  
I will then present a result that extends this correspondence to the non-Gaussian `massless' sine-Gordon field on $R^2$ at $\beta=4\pi$ and massive Dirac fermions.
This is an instance of Coleman's prediction that the `massless' sine-Gordon model and the massive Thirring model are equivalent.
We use this correspondence to show that correlations of the `massless' sine-Gordon model decay exponentially for $\beta=4\pi$.
This is joint work with C. Webb (arXiv:2010.07096).


== March 11, 2021, [https://people.math.rochester.edu/faculty/smkrtchy/ Sevak Mkrtchyan] (Rochester) ==
== March 6, 2025: Alexander Meehan (UW-Madison, Department of Philosophy) ==
'''What conditional probability could (probably) be'''


'''The limit shape of the Leaky Abelian Sandpile Model'''
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.


The leaky abelian sandpile model (Leaky-ASM) is a growth model in which n grains of sand start at the origin in the square lattice and diffuse according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion 1-1/d of its sand. This is a dissipative generalization of the Abelian Sandpile Model, which corresponds to the case d=1.
This talk is based on joint work with Snow Zhang (UC Berkeley).  


We will discuss how, by connecting the model to a certain killed random walk on the square lattice, for any fixed d>1, an explicit limit shape can be computed for the region visited by the sandpile when it stabilizes.
== March 13, 2025: Klara Courteaut (Courant) ==
'''The Coulomb gas on a Jordan arc''' 


We will also discuss the limit shape in the regime when the dissipation parameter d converges to 1 as n grows, as this is related to the ordinary ASM with a modified initial configuration.
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.  


== March 18, 2021, [https://sites.google.com/view/theoassiotis/home Theo Assiotis] (Edinburgh) ==
Based on joint work with Kurt Johansson and Fredrik Viklund.


'''On the joint moments of characteristic polynomials of random unitary matrices'''
== March 20, 2025: Ewain Gwynne (UChicago) ==
'''Random walk reflected off of infinity'''  
I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.


== March 25, 2021, [https://homepages.uc.edu/~brycwz/ Wlodzimierz Bryc] (Cincinnati) ==
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.
'''Fluctuations of particle density  for open ASEP'''


I will review results on fluctuations of particle density for the open Asymmetric Simple Exclusion Process. I will explain the statements and the Laplace transform duality arguments that appear in the proofs.
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.


The talk is based on past and ongoing projects with Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.
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 1, 2021, [https://sites.google.com/view/xiangying-huangs-home-page/home Zoe Huang] (Duke University)  ==
Based on joint work with Jinwoo Sung.
'''Motion by mean curvature in interacting particle systems'''


There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et. al there were two nontrivial stationary distributions.
== March 27, 2025: SPRING BREAK ==
No seminar  


== April 8, 2021, [http://www.math.ucsd.edu/~tiz161/ Tianyi Zheng] (UCSD) ==
== April 3, 2025: Jimme He (OSU) ==
'''Random walks on wreath products and related groups'''
'''Random growth models with half space geometry'''


Random walks on lamplighter groups were first considered by Kaimanovich and Vershik to provide examples of amenable groups with nontrivial Poisson boundary. Such processes can be understood rather explicitly, and provide guidance in the study of random walks on more complicated groups. In this talk we will discuss behavior of random walks on lamplighter groups, their extensions and some related groups which carry a similar semi-direct product structure.
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 15, 2021, [https://stat.wisc.edu/staff/levin-keith/ Keith Levin] (UW-Madison, Statistics) ==
== April 10, 2025: Evan Sorensen (Columbia) ==
'''Network Embeddings and Latent Space Models'''
 
'''Viscous shock fluctuations in KPZ'''


Networks are data structures that describe relations among entities, such as friendships among people in a social network or synapses between neurons in a brain.  The field of statistical network analysis aims to develop network analogues of classical statistical techniques, and latent space models have emerged as the workhorse of this nascent field. Under these models, network formation is driven by unobserved geometric structure, in which each vertex in the network has an associated point in some metric space, called its latent position, that describes the (stochastic) behavior of the vertex in the network. In this talk, I will discuss some of my own work related to latent space models, focusing on 1) estimation of the vertex-level latent positions and 2) generating bootstrap replicates of network data. Throughout the talk, I will make a point to highlight open problems and ongoing projects that are likely to be of interest to probabilists.
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 16, 2021, [http://www.mathjunge.com/ Matthew Junge] (CUNY) <span style="color:red">FRIDAY at 2:25pm, joint with</span> [https://www.math.wisc.edu/wiki/index.php/Applied/ACMS ACMS]  ==
== April 17, 2025: ==
'''Modeling COVID-19 Spread in Universities'''
No seminar 


University policy surrounding COVID-19 often involves big decisions informed by minimal data. Models are a tool to bridge this divide. I will describe some of the work that came out during Summer of 2020 to inform college reopening for Fall 2020. This includes a stochastic, agent-based model on a network for infection spread in residential colleges that I developed alongside a biologist, computer scientist, and group of students [https://arxiv.org/abs/2008.09597]. Time-permitting, I will describe a new project that aims to predict the impact of vaccination on infection spread in urban universities during the Fall 2021 semester. Disclaimer: I self-identify as a "pure" probabilist who typically proves theorems about particle systems [http://www.mathjunge.com/research]. These projects arose from my feeling compelled to help out to the best of my abilities during the height of the pandemic.
== April 24, 2025: William Leeb (University of Minnesota, Twin Cities) ==
TBD 


== April 22, 2021, [https://www.maths.ox.ac.uk/people/benjamin.fehrman Benjamin Fehrman] (Oxford)  ==
== May 1, 2025: Hai-Xiao Wang (UCSD) ==
'''Non-equilibrium fluctuations in interacting particle systems and conservative stochastic PDE'''
TBD
 
Abstract: Interacting particle systems have found diverse applications in mathematics and several related fields, including statistical physics, population dynamics, and machine learning.  We will focus, in particular, on the zero range process and the symmetric simple exclusion process.  The large-scale behavior of these systems is essentially deterministic, and is described in terms of a hydrodynamic limit.  However, the particle process does exhibit large fluctuations away from its mean.  Such deviations, though rare, can have significant consequences---such as a concentration of energy or the appearance of a vacuum---which make them important to understand and simulate.
 
In this talk, which is based on joint work with Benjamin Gess, I will introduce a continuum model for simulating rare events in the zero range and symmetric simple exclusion process.  The model is based on an approximating sequence of stochastic partial differential equations with nonlinear, conservative noise.  The solutions capture to first-order the central limit fluctuations of the particle system, and they correctly simulate rare events in terms of a large deviations principle.
 
== April 29, 2021, [http://www.stats.ox.ac.uk/~martin/ James Martin] (Oxford) ==
 
[[Past Seminars]]

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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