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| | [[Probability | Back to Probability Group]] |
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| = Spring 2021 =
| | * '''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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| <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>
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| <b> IMPORTANT: </b> In Spring 2021 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]
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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 28, 2021, no seminar ==
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| == February 4, 2021, [https://cims.nyu.edu/~hbchen/ Hong-Bin Chen] (Courant Institute, NYU) ==
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| '''Dynamic polymers: invariant measures and ordering by noise'''
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| 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.
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| == February 11, 2021, [https://mathematics.stanford.edu/people/kevin-yang Kevin Yang] (Stanford) == | | == Fall 2025 == |
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| '''Non-stationary fluctuations for some non-integrable models'''
| | <b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b> |
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| 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. | | We usually end for questions at 3:20 PM. |
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| == February 18, 2021, [https://ilyachevyrev.wordpress.com Ilya Chevyrev] (Edinburgh) == | | == September 4, 2025: No seminar == |
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| '''Signature moments to characterize laws of stochastic processes'''
| | == September 11, 2025: David Renfrew (Binghamton U.) == |
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| 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.
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| == February 25, 2021, [https://math.mit.edu/directory/profile.php?pid=2121 Roger Van Peski] (MIT) ==
| | '''Singularities in the spectrum of random block matrices''' |
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| '''Random matrices, random groups, singular values, and symmetric functions'''
| | 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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| 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.
| | == September 18, 2025: JE Paguyo (McMaster U.) == |
| | '''Asymptotic behavior of the hierarchical Pitman-Yor and Dirichlet processes''' |
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| == March 4, 2021, [http://www.statslab.cam.ac.uk/~rb812/ Roland Bauerschmidt] (Cambridge) ==
| | 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. |
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| '''The Coleman correspondence at the free fermion point'''
| | == September 25, 2025: Chris Janjigian (Purdue U.) == |
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| 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.
| | == October 2, 2025: Elliot Paquette (McGill U.) == |
| I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions.
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| 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.
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| This is an instance of Coleman's prediction that the `massless' sine-Gordon model and the massive Thirring model are equivalent.
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| We use this correspondence to show that correlations of the `massless' sine-Gordon model decay exponentially for $\beta=4\pi$.
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| This is joint work with C. Webb (arXiv:2010.07096).
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| == March 11, 2021, [https://people.math.rochester.edu/faculty/smkrtchy/ Sevak Mkrtchyan] (Rochester) == | | == October 9, 2025: No seminar (Midwest Probability Colloquium) == |
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| '''The limit shape of the Leaky Abelian Sandpile Model'''
| | == October 16, 2025: Zachary Selk (Florida State U.) == |
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| 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.
| | '''<br />On the Onsager-Machlup Function for the \Phi^4 Measure''' |
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| 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.
| | 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. |
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| 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.
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| == March 18, 2021, [https://sites.google.com/view/theoassiotis/home Theo Assiotis] (Edinburgh) == | | ==October 23, 2025: Alex Dunlap (Duke U.)== |
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| '''On the joint moments of characteristic polynomials of random unitary matrices'''
| | ==October 30, 2025: Ander Aguirre (UW-Madison)== |
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| 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.
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| == March 25, 2021, [https://homepages.uc.edu/~brycwz/ Wlodzimierz Bryc] (Cincinnati) ==
| | '''Edgeworth expansion and random polynomials''' |
| '''Fluctuations of particle density for open ASEP''' | |
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| 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.
| | ==November 6, 2025: Sudeshna Bhattacharjee (Indian Institute of Science)== |
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| The talk is based on past and ongoing projects with Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.
| | == November 13, 2025: Jiaoyang Huang (U. Penn) == |
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| == April 1, 2021, [https://sites.google.com/view/xiangying-huangs-home-page/home Zoe Huang] (Duke University) == | |
| '''Motion by mean curvature in interacting particle systems'''
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| 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.
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| == April 8, 2021, [http://www.math.ucsd.edu/~tiz161/ Tianyi Zheng] (UCSD) ==
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| '''Random walks on wreath products and related groups'''
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| 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.
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| == April 15, 2021, [https://stat.wisc.edu/staff/levin-keith/ Keith Levin] (UW-Madison, Statistics) ==
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| == 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] ==
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| == April 22, 2021, [https://www.maths.ox.ac.uk/people/benjamin.fehrman Benjamin Fehrman] (Oxford) ==
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| '''Non-equilibrium fluctuations in interacting particle systems and conservative stochastic PDE'''
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| 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.
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| 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.
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| == April 29, 2021, [http://www.stats.ox.ac.uk/~martin/ James Martin] (Oxford) ==
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| [[Past Seminars]]
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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)