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[[Probability | Back to Probability Group]]


= Fall 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 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].
= Spring 2025 =
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>
== September 16, 2021, in person: [https://hanbaeklyu.com/ Hanbaek Lyu] (UW-Madison)  ==


We usually end for questions at 3:20 PM.


'''Scaling limit of soliton statistics of a multicolor box-ball system'''
== January 23, 2025: ==
No seminar 


The box-ball systems (BBS) are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de Veris equation. Probabilistic analysis of BBS is an emerging topic in the field of integrable probability, which often reveals novel connection between the rich integrable structure of BBS and probabilistic phenomena such as phase transition and invariant measures. In this talk, we give an overview on the recent development in scaling limit theory of multicolor BBS with random initial configurations. Our analysis uses various methods such as modified Greene-Kleitman invariants for BBS, circular exclusion processes, Kerov–Kirillov–Reshetikhin bijection, combinatorial R, and Thermodynamic Bethe Ansatz.
== January 30, 2025: Promit Ghosal (UChicago) ==
'''Bridging Theory and Practice in Stein Variational Gradient Descent: Gaussian Approximations, Finite-Particle Rates, and Beyond''' 


== September 23, 2021, no seminar  ==
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) ==
TBD 


== September 30, 2021, in person: [https://mrusskikh.mit.edu/home Marianna Russkikh] (MIT)   ==
== February 13, 2025: Hanbaek Lyu (UW-Madison) ==
'''Large random matrices with given margins''' 


'''Lozenge tilings and the Gaussian free field on a cylinder'''
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. 


We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.
Based on a joint work with Sumit Mukherjee (Columbia) 


== October 7, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://n.ethz.ch/~bdembin/home.html Barbara Dembin] (ETH Zurich)   ==
== February 20, 2025: Mustafa Alper Gunes (Princeton) ==
TBD 


'''The time constant for Bernoulli percolation is Lipschitz continuous strictly above $p_c$'''
== February 27, 2025: Souvik Dhara (Purdue) ==
 
   
We consider the standard model of i.i.d. first passage percolation on $\mathbb Z^d$ given a distribution $G$ on $[0,+\infty]$ ($+\infty$ is allowed). When $G([0,+\infty))>p_c(d)$, it is known that the time constant $\mu_G$ exists. We are interested in the regularity properties of the map $G\mapsto\mu_G$. We study the specific case of distributions of the form $G_p=p\delta_1+(1-p)\delta_\infty$ for $p>p_c(d)$. In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter $p$. We prove that the function $p\mapsto \mu_{G_p}$ is Lipschitz continuous on every interval $[p_0,1]$, where $p_0>p_c(d)$.
'''Propagation of Shocks on Networks: Can Local Information Predict Survival?'''  
This is a joint work with Raphaël Cerf.
 
== October 14, 2021, <span style="color:red">UPDATED FORMAT: </span> [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/wisc.edu/evan-sorensen Evan Sorensen] (UW-Madison) ==
 
'''Busemann functions and semi-infinite geodesics in a semi-discrete space'''
 
In the last 10-15 years, Busemann functions have been a key tool for studying semi-infinite geodesics in planar first and last-passage percolation. We study Busemann functions in the semi-discrete Brownian last-passage percolation (BLPP) model and use these to derive geometric properties of the full collection of semi-infinite geodesics in BLPP. This includes a characterization of uniqueness and coalescence of semi-infinite geodesics across all asymptotic directions. To deal with the uncountable set of points in BLPP, we develop new methods of proof and uncover new phenomena, compared to discrete models. For example, for each asymptotic direction, there exists a random countable set of initial points out of which there exist two semi-infinite geodesics in that direction. Further, there exists a random set of points, of Hausdorff dimension ½, out of which, for some random direction, there are two semi-infinite geodesics that split from the initial point and never come back together. We derive these results by studying variational problems for Brownian motion with drift.
 
== October 21, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [http://stat.columbia.edu/~sumitm/ Sumit Mukherjee] (Columbia)    ==
 
'''Fluctuations in Mean Field Ising Models'''
 
We study fluctuations of the magnetization (average of spins) in an Ising model on a sequence of "well-connected" approximately $d_n$ regular graphs on $n$ vertices. We show that if $d_n\gg n^{1/2}$,  then the fluctuations are universal, and same as that of the Curie–Weiss model, in the entire ferromagnetic parameter regime. We then give a counterexample to show that $d_n\gg n^{1/2}$ is actually tight, in the sense that the limiting distribution changes if $d_n\sim n^{1/2}$ except in the high temperature regime. By refining our argument, we show that in the high temperature regime universality holds for $d_n\gg n^{1/3}$. As a by-product of our proof technique, we prove rates of convergence, as well as exponential concentration for the sum of spins, and tight estimates for several statistics of interest.
 
This is based on joint work with Nabarun Deb at Columbia University.
 
== October 28, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www-users.cse.umn.edu/~wkchen/ Wei-Kuo Chen] (Minnesota)    ==
 
'''Grothendieck $L_p$ problem for Gaussian matrices'''
 
The Grothendieck $L_p$ problem is defined as an optimization problem that maximizes the quadratic form of a Gaussian matrix over the unit $L_p$ ball. The $p=2$ case corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble, while for $p=\infty$ this problem is known as the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In this talk, I will describe the limit of this optimization problem for general $p$ and discuss some results on the behavior of the near optimizers along with some open problems. This is based on a joint work with Arnab Sen.
 
== November 4, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://mathematics.stanford.edu/people/mackenzie-simper Mackenzie Simper] (Stanford)  ==
 
'''Double Cosets, Mallows Measure, and a Transvections Markov Chain'''


If $G = GL_n(\mathbb{F}_q)$ and $B$ is the subgroup of lower triangular matrices, then the $B\backslash G/B$ double cosets are indexed by permutations $S_n$. This is the famous Bruhat decomposition, closely related to the LU decomposition of a matrix. The Markov chain on $G$ generated by random transvections – matrices which fix a hyperplane – induces a Markov chain on $S_n$ with the Mallows measure as stationary distribution. We characterize this process, study the mixing time, and discuss the connection with the number of pivoting steps needed in Gaussian elimination. This is joint work with Persi Diaconis and Arun Ram.
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.


== November 11, 2021, in person: [http://web.lfzhang.com/ Lingfu Zhang] (Princeton)   ==
== March 6, 2025: Alexander Meehan (UW-Madison, Department of Philosophy) ==
'''What conditional probability could (probably) be'''


'''Shift-invariance of the colored TASEP and random sorting networks'''
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.


In this talk, I will introduce a new shift-invariance property of the colored TASEP. It is in a similar spirit as recent results for the six-vertex models (by Borodin-Gorin-Wheeler and Galashin), but its proof is via non-algebraic arguments. This new shift-invariance is applied to prove a conjectured distributional equality between the classical exponential Last Passage Percolation model and the oriented swap process (OSP). The OSP is a model for a random sorting network, with N particles labeled $1,\dots,N$ performing successive adjacent swaps at random times until they reach the reverse configuration $N,\dots,1$. Our distributional equality implies new asymptotic results about the OSP, some of which are in connection with the Airy sheet.
This talk is based on joint work with Snow Zhang (UC Berkeley).  


== November 18, 2021, in person: [http://www.ilt.kharkov.ua/bvi/structure/depart_e/d24/mariya_shcherbina-cv.htm Mariya Shcherbina]: (Kharkov) ==
== March 13, 2025: Klara Courteaut (Courant) ==
'''Supersymmetric approach to the deformed Ginibre ensemble.'''
TBD 


We consider  non Hermitian random matrices of the form
== March 20, 2025: Ewain Gwynne (UChicago) ==
$H=A+H_0$, where $A$ is a rather general $n\times n$ matrix (Hermitian
TBD 
or non-Hermitian)
independent of $H_0$, and $H_0$ is a standard Ginibre  matrix.
It is known that under some reasonable conditions the limiting spectrum
of $H$ makes
some domain $D$ with a smooth boundary $\Gamma$. We apply the
supersymmetric approach
to study the behavior of the smallest singular value of the matrix
$(H-z)$  if $z\in D$
but $dist(z,\Gamma)\sim N^{-1/2}$.


== November 25, 2021, no seminar  ==
== March 27, 2025: SPRING BREAK ==
No seminar   


== April 3, 2025: Jimme He (OSU) ==
TBD 


== December 2, 2021, in person: [http://math.uchicago.edu/~xuanw/ Xuan Wu] (Chicago) ==
== April 10, 2025: Evan Sorensen (Columbia) ==
TBD 


== April 17, 2025: ==
TBD 


== December 9, 2021, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www.maths.dur.ac.uk/users/sunil.chhita/ Sunil Chhita] (Durham) ==
== April 24, 2025: William Leep (University of Minnesota, Twin Cities) ==
TBD 


'''GOE Fluctuations for the maximum of the top path in ASMs'''
== May 1, 2025: ==
 
No seminar
The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter Δ. When Δ=0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all Δ, there has been very little progress in understanding its statistics in the scaling limit for other values. In this talk, we focus on the six-vertex model with domain wall boundary conditions at Δ=1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We report that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy-Widom distribution after appropriate rescaling.  This talk is based on joint work with Arvind Ayyer and Kurt Johansson.
 
 
[[Past Seminars]]

Latest revision as of 07:27, 28 January 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)

TBD

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)

TBD

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)

TBD

March 20, 2025: Ewain Gwynne (UChicago)

TBD

March 27, 2025: SPRING BREAK

No seminar

April 3, 2025: Jimme He (OSU)

TBD

April 10, 2025: Evan Sorensen (Columbia)

TBD

April 17, 2025:

TBD

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

TBD

May 1, 2025:

No seminar

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