Probability Seminar: Difference between revisions

From UW-Math Wiki
Jump to navigation Jump to search
VisualWikitext
No edit summary
 
(135 intermediate revisions by 9 users not shown)
Line 1: Line 1:
__NOTOC__
__NOTOC__
[[Probability | Back to Probability Group]]


= Spring 2022 =
* '''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 at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>
[[Past Seminars]]
 
We  usually end for questions at 3:20 PM.
 
[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]
 
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].
 
 
== February 3, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://zhipengliu.ku.edu/ Zhipeng Liu] (University of Kansas)    ==
 
'''One-point distribution of the geodesic in directed last passage percolation'''


In the recent twenty years, there has been a huge development in understanding the universal law behind a family of 2d random growth models, the so-called Kardar-Parisi-Zhang (KPZ) universality class. Especially, limiting distributions of the height functions are identified for a number of models in this class. Different from the height functions, the geodesics of these models are much less understood. There were studies on the qualitative properties of the geodesics in the KPZ universality class very recently,  but the precise limiting distributions of the geodesic locations remained unknown.


In this talk, we will discuss our recent results on the one-point distribution of the geodesic of a representative model in the KPZ universality class, the directed last passage percolation with iid exponential weights. We will give the explicit formula of the one-point distribution of the geodesic location joint with the last passage times, and its limit when the parameters go to infinity under the KPZ scaling. The limiting distribution is believed to be universal for all the models in the KPZ universality class. We will further give some applications of our formulas.


== February 10, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://jscalvert.github.io/ Jacob Calvert] (U.C. Berkeley)    ==
= Spring 2025 =
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>


'''Harmonic activation and transport'''
We usually end for questions at 3:20 PM.


Models of Laplacian growth, such as diffusion-limited aggregation (DLA), describe interfaces which move in proportion to harmonic measure. I will introduce a model, called harmonic activation and transport (HAT), in which a finite subset of Z^2 is rearranged according to harmonic measure. HAT exhibits a phenomenon called collapse, whereby the diameter of the set is reduced to its logarithm over a number of steps comparable to this logarithm. I will describe how collapse can be used to prove the existence of the stationary distribution of HAT, which is supported on a class of sets viewed up to translation. Lastly, I will discuss the problem of quantifying the least positive harmonic measure as a function of set cardinality, which arises in the study of HAT, and a partial resolution of which rules out predictions about DLA from the physics literature. Based on joint work with Shirshendu Ganguly and Alan Hammond.
== January 23, 2025: ==
No seminar 


== February 17, 2022, in person: [https://sites.math.northwestern.edu/~kivimae/ Pax Kivimae] (Northwestern University)   ==  
== January 30, 2025: Promit Ghosal (UChicago) ==
'''Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond''' 


'''The Ground-State Energy and Concentration of Complexity in Spherical Bipartite 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.


Bipartite spin glass models have been gaining popularity in the study of glassy systems with distinct interacting species. Recently, the annealed complexity of the pure spherical bipartite model was obtained by B. McKenna. In this talk, I will explain how to show that the low-lying complexity actually concentrates around this value, and how from this one can obtain a formula for the ground-state energy.
== February 6, 2025: Subhabrata Sen (Harvard) ==
'''Community detection on multi-view networks''' 


== February 24, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [http://math.uchicago.edu/~lbenigni/ Lucas Benigni] (University of Chicago)  ==
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.


'''Optimal delocalization for generalized Wigner matrices'''
This is based on joint work with Xiaodong Yang and Buyu Lin (Harvard University). 


We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.
== February 13, 2025: Hanbaek Lyu (UW-Madison) ==
'''Large random matrices with given margins''' 


== March 3, 2022, in person: [https://math.wisc.edu/staff/keating-david/ David Keating] (UW-Madison)  ==
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. 


'''$k$-tilings of the Aztec diamond'''
Based on a joint work with Sumit Mukherjee (Columbia) 


We study $k$-tilings ($k$-tuples of domino tilings) of the Aztec diamond of rank $m$.  We assign a weight to each $k$-tiling, depending on the number of vertical dominos and also on the number of "interactions" between the different tilings.  We will compute the generating polynomials of the $k$-tilings by relating them to an integrable colored vertex model. We will then prove some combinatorial results about $k$-tilings in certain limits of the interaction strength.
== February 20, 2025: Mustafa Alper Gunes (Princeton) ==
'''Characteristic Polynomials of Random Matrices, Exchangeable Arrays & Painlevé Equations'''  


== March 10, 2022, in person: [https://qiangwu2.github.io/martingale/ Qiang Wu] (University of Illinois Urbana-Champaign)  ==
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.


'''Mean field spin glass models under weak external field'''
== February 27, 2025: Souvik Dhara (Purdue) ==
'''Propagation of Shocks on Networks: Can Local Information Predict Survival?'''  


We study the fluctuation and limiting distribution of free energy in mean-field spin glass models with Ising spins under weak external fields. We prove that at high temperature, there are three regimes concerning the strength of external field $h \approx \rho N^{-\alpha}$ with $\rho,\alpha\in (0,\infty)$. In the super-critical regime $\alpha < 1/4$, the variance of the log-partition function is $\approx N^{1-4\alpha}$. In the critical regime $\alpha = 1/4$, the fluctuation is of constant order but depends on $\rho$. Whereas, in the sub-critical regime $\alpha>1/4$, the variance is $\Theta(1)$ and does not depend on $\rho$. We explicitly express the asymptotic mean and variance in all three regimes and prove Gaussian central limit theorems. Our proofs mainly follow two approaches. One utilizes quadratic coupling and Guerra's interpolation scheme for Gaussian disorder, extending to many other spin glass models. However, this approach can prove the CLT only at very high temperatures. The other one is a cluster-based approach for general symmetric disorders, first used in the seminal work of Aizenman, Lebowitz, and Ruelle (Comm. Math. Phys. 112 (1987), no. 1, 3-20) for the zero external field case. It was believed that this approach does not work if the external field is present. We show that if the external field is present but not too strong, it still works with a new cluster structure. In particular, we prove the CLT up to the critical temperature in the Sherrington-Kirkpatrick (SK) model when $\alpha \ge 1/4$. We further address the generality of this cluster-based approach. Specifically, we give similar results for the multi-species SK model and diluted SK model. Based on joint work with Partha S. Dey.
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 24, 2022, in person: [http://math.columbia.edu/~sayan/ Sayan Das] (Columbia University)   ==  
== March 6, 2025: Alexander Meehan (UW-Madison, Department of Philosophy) ==
'''What conditional probability could (probably) be'''


'''Path properties of the KPZ Equation and related polymers'''
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 KPZ equation is a fundamental stochastic PDE that can be viewed as the log-partition function of continuum directed random polymer (CDRP). In this talk, we will first focus on the fractal properties of the tall peaks of the KPZ equation. This is based on separate joint works with Li-Cheng Tsai and Promit Ghosal. In the second part of the talk, we will study the KPZ equation through the lens of polymers. In particular, we will discuss localization aspects of CDRP that will shed light on certain properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum. This is based on joint work with Weitao Zhu.  
This talk is based on joint work with Snow Zhang (UC Berkeley).  


== March 31, 2022, in person: [http://willperkins.org/ Will Perkins] (University of Illinois Chicago)   ==  
== March 13, 2025: Klara Courteaut (Courant) ==
'''The Coulomb gas on a Jordan arc''' 


'''Potential-weighted connective constants and uniqueness of Gibbs measures'''
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. 


Classical gases (or Gibbs point processes) are models of gases or fluids, with particles interacting in the continuum via a density against background Poisson processes.  The major questions about these models are about phase transitions, and while these models have been studied in mathematics for over 70 years, some of the most fundamental questions remain open. After giving some background on these models, I will describe a new method for proving absence of phase transition (uniqueness of infinite volume Gibbs measures) in the low-density regime of processes interacting via a repulsive pair potential.  The method involves defining a new quantity, the "potential-weighted connective constant", and is motivated by techniques from computer science.  Joint work with Marcus Michelen.
Based on joint work with Kurt Johansson and Fredrik Viklund.   


== April 7, 2022, [in person and on https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/view/eliza-oreilly/home Eliza O'Reilly] (Caltech)   ==  
== March 20, 2025: Ewain Gwynne (UChicago) ==
'''Random walk reflected off of infinity''' 


'''Stochastic Geometry for Machine Learning'''
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 Mondrian process is a stochastic process that produces a recursive partition of space with random axis-aligned cuts. It has been used in machine learning to build random forests and Laplace kernel approximations.  The construction allows for efficient online algorithms, but the restriction to axis-aligned cuts does not capture dependencies between features. By viewing the Mondrian as a special case of the stable under iterated (STIT) process in stochastic geometry, we resolve open questions about the generalization of cut directions. We utilize the theory of stationary random tessellations to show that STIT random forests achieve minimax rates for Lipschitz and $C^2$ functions and STIT random features approximate a large class of stationary kernels. This work opens many new questions at the intersection of stochastic geometry and machine learning. Based on joint work with Ngoc Mai Tran.
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.


== April 14, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://cmps.ok.ubc.ca/about/contact/eric-foxall/ Eric Foxall] (UBC-Okanagan)   ==
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 $c > 95/4$).


'''Bifurcation theory of density-dependent Markov chains, well-mixed stochastic population models, and diffusively perturbed dynamical systems'''
Based on joint work with Jinwoo Sung.


Abstract: A common choice for modeling a finite, interacting population of individuals is to specify a variable system size parameter $N$, and to otherwise assume that interactions are well-mixed. When the model is cast in continuous time, has finitely many types, and is otherwise fairly simple, the vector $X$ of population sizes is a density-dependent Markov chain (DDMC), and the population density vector $\frac{X}{N}$ can be roughly seen as a diffusive perturbation of a deterministic system, with noise parameter $\varepsilon = 1/\sqrt{N}$. The law of large numbers and central limit theorem of these models has been known since the work of Thomas Kurtz in the 1970s. Here, we study bifurcations of parametrized models of this type, as a function of $\varepsilon$. We introduce the notion of limit scales to understand the shape of fluctuations in and around the bifurcation point, and give a three-step framework for finding them. The result is an enhanced bifurcation diagram that encodes this information with the help of a few well-chosen functions. We develop the theory in detail for one-dimensional models and illustrate it for simple bifurcation types including transcritical, saddle-node, pitchfork, and the Hopf bifurcation.
== March 27, 2025: SPRING BREAK ==
No seminar 


== April 21, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: Hugo Falconet (NYU)   ==  
== April 3, 2025: Jimme He (OSU) ==
'''Random growth models with half space geometry''' 


'''TBA'''
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 28, 2022, in person: [https://www.ias.edu/scholars/amol-aggarwal Amol Aggarwal] (Columbia/IAS)   ==  
== April 10, 2025: Evan Sorensen (Columbia) ==
 
'''Viscous shock fluctuations in KPZ''' 


'''TBA'''
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.   


== May 5, 2022, in person: [https://people.math.gatech.edu/~dharper40/ David Harper] (Georgia Tech)  ==  
== April 17, 2025: ==
No seminar 


'''TBA'''
== April 24, 2025: William Leeb (University of Minnesota, Twin Cities) ==
TBD 


[[Past Seminars]]
== 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

Retrieved from "https://wiki.math.wisc.edu/index.php?title=Probability_Seminar&oldid=28356"