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| [[Probability | Back to Probability Group]] | | [[Probability | Back to Probability Group]] |
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| | * '''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 |
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| [[Past Seminars]] | | [[Past Seminars]] |
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| = Spring 2023 = | | == Fall 2025 == |
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| <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> |
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| We usually end for questions at 3:20 PM. | | We usually end for questions at 3:20 PM. |
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| [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]
| | == September 4, 2025: No seminar == |
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| 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].
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| == January 26, 2023, in person: [https://sites.google.com/wisc.edu/evan-sorensen?pli=1 Evan Sorensen] (UW-Madison) == | |
| '''The stationary horizon as a universal object for KPZ models'''
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| The last 5-10 years has seen remarkable progress in constructing the central objects of the KPZ universality class, namely the KPZ fixed point and directed landscape. In this talk, I will discuss a third central object known as the stationary horizon (SH). The SH is a coupling of Brownian motions with drifts, indexed by the real line, and it describes the unique coupled invariant measures for the directed landscape. I will talk about how the SH appears as the scaling limit of several models, including Busemann processes in last-passage percolation and the TASEP speed process. I will also discuss how the SH helps to describe the collection of infinite geodesics in all directions for the directed landscape. Based on joint work with Timo Seppäläinen and Ofer Busani.
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| == February 2, 2023, in person: [https://mathjinsukim.com/ Jinsu Kim] (POSTECH) ==
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| '''Fast and slow mixing of continuous-time Markov chains with polynomial rates'''
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| Continuous-time Markov chains on infinite positive integer grids with polynomial rates are often used in modeling queuing systems, molecular counts of small-size biological systems, etc. In this talk, we will discuss continuous-time Markov chains that admit either fast or slow mixing behaviors. For a positive recurrent continuous-time Markov chain, the convergence rate to its stationary distribution is typically investigated with the Lyapunov function method and canonical path method. Recently, we discovered examples that do not lend themselves easily to analysis via those two methods but are shown to have either fast mixing or slow mixing with our new technique. The main ideas of the new methodologies are presented in this talk along with their applications to stochastic biochemical reaction network theory.
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| == February 9, 2023, in person: [https://www.math.tamu.edu/~jkuan/ Jeffrey Kuan] (Texas A&M) ==
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| '''Shift invariance for the multi-species q-TAZRP on the infinite line'''
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| We prove a shift--invariance for the multi-species q-TAZRP (totally asymmetric zero range process) on the infinite line. Similar-looking results had appeared in works by [Borodin-Gorin-Wheeler] and [Galashin], using integrability, but are on the quadrant. The proof in this talk relies instead on a combinatorial approach, in which the state space is generalized to a poset, and the totally asymmetric process is generalized to a monotone process on a poset. The continuous-time process is decomposed into its discrete embedded Markov chain and its exponential holding times, and the shift-invariance is proved using explicit contour integral formulas. Open problems about multi-species ASEP will be discussed as well.
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| == February 16, 2023, in person: [http://math.columbia.edu/~milind/ Milind Hegde] (Columbia) ==
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| '''Understanding the upper tail behaviour of the KPZ equation via the tangent method'''
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| The Kardar-Parisi-Zhang (KPZ) equation is a canonical non-linear stochastic PDE believed to describe the evolution of a large number of planar stochastic growth models which make up the KPZ universality class. A particularly important observable is the one-point distribution of its analogue of the fundamental solution, which has featured in much of its recent study. However, in spite of significant recent progress relying on explicit formulas, a sharp understanding of its upper tail behaviour has remained out of reach. In this talk we will discuss a geometric approach, related to the tangent method introduced by Colomo-Sportiello and rigorously implemented by Aggarwal for the six-vertex model. The approach utilizes a Gibbs resampling property of the KPZ equation and yields a sharp understanding for a large class of initial data.
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| == February 23, 2023, in person: [https://sites.math.rutgers.edu/~sc2518/ Swee Hong Chan] (Rutgers) ==
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| '''Log-concavity and cross product inequalities in order theory'''
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| Given a finite poset that is not completely ordered, is it always possible find two elements x and y, such that the probability that x is less than y in the random linear extension of the poset, is bounded away from 0 and 1? Kahn-Saks gave an affirmative answer and showed that this probability falls between 3/11 (0.273) and 8/11 (0.727). The currently best known bound is 0.276 and 0.724 by Brightwell-Felsner-Trotter, and it is believed that the optimal bound should be 1/3 and 2/3, also known as the 1/3-2/3 Conjecture. Most notably, log-concave and cross product inequalities played the central role in deriving both bounds. In this talk we will discuss various generalizations of these results together with related open problems. This talk is joint work with Igor Pak and Greta Panova, and is intended for the general audience.
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| == March 2, 2023, in person: Max Hill (UW-Madison) ==
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| '''On the Effect of Intralocus Recombination on Triplet-Based Species Tree Estimation'''
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| My talk will introduce some key topics in mathematical phylogenetics and is intended to be accessible for those not familiar with the field. I will discuss joint work with Sebastien Roch on the subject of species tree estimation from multiple loci subject to intralocus recombination. The focus is on R*, a summary coalescent-based method using rooted triplets. I will present a result showing how intralocus recombination can give rise to an "inconsistency zone," in which correct inference using R* is not assured even in the limit of infinite amount of data.
| | == September 11, 2025: David Renfrew (Binghamton U.) == |
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| == March 9, 2023, in person: [https://math.uchicago.edu/~xuanw/ Xuan Wu] (U. Chicago) ==
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| '''From the KPZ equation to the directed landscape'''
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| This talk presents the convergence of the KPZ equation to the directed landscape, which is the central object in the KPZ universality class. This convergence result is the first to the directed landscape among the positive temperature models.
| | '''Singularities in the spectrum of random block matrices''' |
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| == March 23, 2023, in person: Jiaming Xu (UW-Madison) ==
| | 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. |
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| '''Rectangular Matrix addition in low and high temperatures''' | | == September 18, 2025: JE Paguyo (McMaster U.) == |
| | '''Asymptotic behavior of the hierarchical Pitman-Yor and Dirichlet processes''' |
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| We study the addition of two <math>{\scriptsize M \times N}</math> rectangular random matrices with certain
| | 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. |
| invariant distributions in two limit regimes, where the parameter <math>{\scriptsize \beta}</math> (inverse temperature) goes to infinity and zero. In low temperature regime the random singular values of the sum concentrate at deterministic points, while in high temperature regime we obtain a Law of Large Numbers of the empirical measures. Our proof uses the so-called type BC Bessel function as characteristic function of rectangular matrices, and through the analysis of this function we introduce a new family of cumulants, that linearize the addition in high temperature limit, and
| | 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. |
| degenerate to the classical or free cumulants in special cases.
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| == March 30, 2023, in person: [http://www.math.toronto.edu/balint/ Bálint Virág] (Toronto) == | | == September 25, 2025: Chris Janjigian (Purdue U.) == |
| '''The planar stochastic heat equation and the directed landscape'''
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| The planar stochastic heat equation describes heat flow or random polymers on an inhomogeneous surface. It is a finite-temperature version of planar first passage percolation such as the Eden growth model. It is the first model with plane symmetries for which we can show convergence to the directed landscape. The methods use a Skorokhod integral representation and Gaussian multiplicative chaos on path space.
| | == October 2, 2025: Elliot Paquette (McGill U.) == |
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| Joint work with Jeremy Quastel and Alejandro Ramirez.
| | == October 9, 2025: No seminar (Midwest Probability Colloquium) == |
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| == April 6, 2023, in person: [https://shankarbhamidi.web.unc.edu/ Shankar Bhamidi] (UNC-Chapel Hill) == | | == October 16, 2025: Zachary Selk (Florida State U.) == |
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| '''Disorder models for random graphs, Erdos’s leader problem, and power of limited choice models for network evolution''' | | '''<br />On the Onsager-Machlup Function for the \Phi^4 Measure''' |
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| First passage percolation, and more generally the study of diffusion of material through disordered systems is a fundamental area in probabilistic combinatorics with a vast body of work especially in the context of spatial systems.
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| The goal of this talk is to survey a slightly different setting for such questions namely the more “mean-field” setting of random graph models. We will describe the state of the art of this field, with the final goal of describing one of the main conjectures in this area namely the conjectured scaling limit of the minimal spanning tree and its dependence on the degree exponent of the corresponding network model. We will describe recent progress in this area, its connection to questions in dynamic network models, in particular Erdos’s leader problem for the identity of the maximal component for critical random graphs, and the intuition for understanding the evolution of maximal components through the critical scaling window from a different area of probabilistic combinatorics, namely the study of limited choice models for network evolution.
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| == April 13, 2023, in person: [http://www.bricehuang.com/index.html Brice Huang] (MIT) ==
| | 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. |
| '''Algorithmic Threshold for Multi-Species Spherical Spin Glasses'''
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| This talk focuses on optimizing the random and non-convex Hamiltonians of spherical spin glasses with multiple species. Our main result identifies the best possible value ALG achievable by class of Lipschitz algorithms and gives a matching algorithm in this class based on approximate message passing. The threshold ALG is given by a certain variational problem, which surprisingly may possess multiple optimizers.
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| Our hardness result is proved using the Branching OGP introduced in our previous work [H-Sellke 21] to identify ALG for single-species spin glasses. This and all other OGPs for spin glasses have been proved using Guerra's interpolation method. We introduce a new method to prove the Branching OGP which is both simpler and more robust. It works even for models in which the true maximum value of the objective function remains unknown.
| | ==October 23, 2025: Alex Dunlap (Duke U.)== |
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| Based on joint work with Mark Sellke.
| | ==October 30, 2025: Ander Aguirre (UW-Madison)== |
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| == April 20, 2023, in person: [http://www.math.columbia.edu/~remy/ Guillaume Remy] (IAS) ==
| | '''Edgeworth expansion and random polynomials''' |
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| == April 27, 2023, in person: [http://www.math.tau.ac.il/~peledron/ Ron Peled] (Tel Aviv/IAS) == | | ==November 6, 2025: Sudeshna Bhattacharjee (Indian Institute of Science)== |
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| == May 4, 2023, in person: [https://www.asc.ohio-state.edu/sivakoff.2// David Sivakoff] (Ohio State) == | | == November 13, 2025: Jiaoyang Huang (U. Penn) == |
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)