Difference between revisions of "Probability Seminar"

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__NOTOC__
 
__NOTOC__
  
= Fall 2020 =
+
= Spring 2021 =
  
 
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.  
 
<b>Thursdays in 901 Van Vleck Hall at 2:30 PM</b>, unless otherwise noted.  
 
<b>We  usually end for questions at 3:20 PM.</b>
 
<b>We  usually end for questions at 3:20 PM.</b>
  
<b> IMPORTANT: </b> In Fall 2020 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]
+
<b> IMPORTANT: </b> In Spring 2021 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]
  
 
If you would like to sign up for the email list to receive seminar announcements then please join [https://groups.google.com/a/g-groups.wisc.edu/forum/#!forum/probsem our group].
 
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].
 
   
 
   
== September 17, 2020, [https://www.math.tamu.edu/~bhanin/ Boris Hanin] (Princeton and Texas A&M) ==
+
== January 28, 2021, no seminar  ==
  
'''Pre-Talk: (1:00pm)'''
+
== February 4, 2021, [https://cims.nyu.edu/~hbchen/ Hong-Bin Chen] (Courant Institute, NYU) ==
  
'''Neural Networks for Probabilists'''
+
'''Dynamic polymers: invariant measures and ordering by noise'''
  
Deep neural networks are a centerpiece in modern machine learning. They are also fascinating probabilistic models, about which much remains unclear. In this pre-talk I will define neural networks, explain how they are used in practice, and give a survey of the big theoretical questions they have raised. If time permits, I will also explain how neural networks are related to a variety of classical areas in probability and mathematical physics, including random matrix theory, optimal transport, and combinatorics of hyperplane arrangements.
+
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.
  
'''Talk: (2:30pm)'''
+
== February 11, 2021, [https://mathematics.stanford.edu/people/kevin-yang Kevin Yang] (Stanford) ==
  
'''Effective Theory of Deep Neural Networks'''  
+
'''Non-stationary fluctuations for some non-integrable models'''
  
Deep neural networks are often considered to be complicated "black boxes," for which a full systematic analysis is not only out of reach but also impossible. In this talk, which is based on ongoing joint work with Sho Yaida and Daniel Adam Roberts, I will make the opposite claim. Namely, that deep neural networks with random weights and biases are exactly solvable models. Our approach applies to networks at finite width n and large depth L, the regime in which they are used in practice. A key point will be the emergence of a notion of "criticality," which involves a finetuning of model parameters (weight and bias variances). At criticality, neural networks are particularly well-behaved but still exhibit a tension between large values for n and L, with large values of n tending to make neural networks more like Gaussian processes and large values of L amplifying higher cumulants. Our analysis at initialization has many consequences also for networks during after training, which I will discuss if time permits.
+
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.
  
== September 24, 2020, [https://people.ucd.ie/neil.oconnell Neil O'Connell] (Dublin) ==
+
== February 18, 2021, [https://ilyachevyrev.wordpress.com Ilya Chevyrev] (Edinburgh) ==
  
'''Some new perspectives on moments of random matrices'''
+
'''Signature moments to characterize laws of stochastic processes'''
  
The study of `moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.
+
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.
  
== October 1, 2020, [https://marcusmichelen.org/ Marcus Michelen] (UIC) ==
+
== February 25, 2021, [https://math.mit.edu/directory/profile.php?pid=2121 Roger Van Peski] (MIT) ==
  
'''Roots of random polynomials near the unit circle'''
+
'''Random matrices, random groups, singular values, and symmetric functions'''
  
It is a well-known (but perhaps surprising) fact that a polynomial with independent random coefficients has most of its roots very close to the unit circle. Using a probabilistic perspective, we understand the behavior of roots of random polynomials exceptionally close to the unit circle and prove several limit theorems; these results resolve several conjectures of Shepp and Vanderbei. We will also discuss how our techniques provide a heuristic, probabilistic explanation for why random polynomials tend to have most roots near the unit circle.  Based on joint work with Julian Sahasrabudhe.
+
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.
  
== October 8, 2020, [http://sites.harvard.edu/~sus977/index.html Subhabrata Sen] (Harvard) ==
+
== March 4, 2021, [http://www.statslab.cam.ac.uk/~rb812/ Roland Bauerschmidt] (Cambridge) ==
  
'''Large deviations for dense random graphs: beyond mean-field'''
+
'''The Coleman correspondence at the free fermion point'''
  
In a seminal paper, Chatterjee and Varadhan derived an Erdős-Rényi random graph, viewed as a random graphon. This directly provides LDPs for continuous functionals such as subgraph counts, spectral norms, etc. In contrast, very little is understood about this problem if the underlying random graph is inhomogeneous or constrained.
+
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.
 +
I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions.
 +
I will then present a result that extends this correspondence to the non-Gaussian `massless' sine-Gordon field on $R^2$ at $\beta=4\pi$ and massive Dirac fermions.
 +
This is an instance of Coleman's prediction that the `massless' sine-Gordon model and the massive Thirring model are equivalent.  
 +
We use this correspondence to show that correlations of the `massless' sine-Gordon model decay exponentially for $\beta=4\pi$.
 +
This is joint work with C. Webb (arXiv:2010.07096).
  
In this talk, we will explore large deviations for dense random graphs, beyond the “mean-field” setting. In particular, we will study large deviations for uniform random graphs with given degrees, and a family of dense block model
+
== March 11, 2021, [https://people.math.rochester.edu/faculty/smkrtchy/ Sevak Mkrtchyan] (Rochester)  ==
random graphs. We will establish the LDP in each case, and identify the rate function. In the block model setting, we will use this LDP to study the upper tail problem for homomorphism densities of regular sub-graphs. Our results establish that this problem exhibits a symmetry/symmetry-breaking transition, similar to one observed for Erdős-Rényi random graphs.
 
  
Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.
+
'''The limit shape of the Leaky Abelian Sandpile Model'''
  
== October 15, 2020, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
+
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.
  
Title: '''Concentration in integrable polymer models'''
+
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.
  
I will discuss a general method, applicable to all known integrable stationary polymer models, to obtain nearly optimal bounds on the
+
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.
central moments of the partition function and the occupation lengths for each level of the polymer system. The method was developed
 
for the O'Connell-Yor polymer, but was subsequently extended to discrete integrable polymers. As an application, we obtain
 
localization of the OY polymer paths along a straight line on the scale O(n^{2/3+o(1)}).  
 
  
Joint work with Christian Noack.
+
== March 18, 2021, [https://sites.google.com/view/theoassiotis/home Theo Assiotis] (Edinburgh) ==
  
==October 22, 2020, [http://www.math.toronto.edu/balint/ Balint Virag] (Toronto) ==
+
'''On the joint moments of characteristic polynomials of random unitary matrices'''
 
+
Title: '''The heat and the landscape'''
+
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.
 
 
Abstract: The directed landscape is the conjectured universal scaling limit of the
 
most common random planar metrics. Examples are planar first passage
 
percolation, directed last passage percolation, distances in percolation
 
clusters, random polymer models, and exclusion processes. The limit laws of distances of objects are given by the KPZ fixed point.
 
 
 
We show that the KPZ fixed point is characterized by the Baik Ben-Arous
 
Peche statistics well-known from random matrix theory.
 
 
 
This provides a general and elementary method for showing convergence to
 
the KPZ fixed point. We apply this method to two models related to
 
random heat flow: the O'Connell-Yor polymer and the KPZ equation.
 
 
 
Note: there will be a follow-up talk with details about the proofs at 11am, Friday, October 23.
 
 
 
==October 29, 2020, [https://www.math.wisc.edu/node/80 Yun Li] (UW-Madison) ==
 
 
 
Title: '''Operator level hard-to-soft transition for β-ensembles'''
 
 
 
Abstract: It was shown that the soft and hard edge scaling limits of β-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. By tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit. In this talk, I will present the corresponding limit on the level of the operators. This talk is based on joint work with Laure Dumaz and Benedek Valkó.
 
 
 
== November 5, 2020, [http://sayan.web.unc.edu/ Sayan Banerjee] (UNC at Chapel Hill) ==
 
 
 
Title: '''Persistence and root detection algorithms in growing networks'''
 
  
Abstract: Motivated by questions in Network Archaeology, we investigate statistics of dynamic networks
+
== March 25, 2021, [https://homepages.uc.edu/~brycwz/ Wlodzimierz Bryc] (Cincinnati) ==
that are ''persistent'', that is, they fixate almost surely after some random time as the network grows. We
+
'''Fluctuations of particle density  for open ASEP'''
consider ''generalized attachment models'' of network growth where at each time $n$, an incoming vertex
 
attaches itself to the network through $m_n$ edges attached one-by-one to existing vertices with probability
 
proportional to an ''arbitrary function'' $f$ of their degree. We identify the class of attachment functions $f$ for
 
which the ''maximal degree vertex'' persists and obtain asymptotics for its index when it does not. We also
 
show that for tree networks, the ''centroid'' of the tree persists and use it to device polynomial time root
 
finding algorithms and quantify their efficacy. Our methods rely on an interplay between dynamic
 
random networks and their continuous time embeddings.
 
  
This is joint work with Shankar Bhamidi.
+
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 12, 2020, [https://cims.nyu.edu/~ajd594/ Alexander Dunlap] (NYU Courant Institute) ==
+
The talk is based on past and ongoing projects with  Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.
  
Title: '''A forward-backward SDE from the 2D nonlinear stochastic heat equation'''
+
== 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'''
  
Abstract: I will discuss a two-dimensional stochastic heat equation in the weak noise regime with a nonlinear noise strength. I will explain how pointwise statistics of solutions to this equation, as the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor, can be related to a forward-backward stochastic differential equation (FBSDE) depending on the nonlinearity. In the linear case, the FBSDE can be explicitly solved and we recover results of Caravenna, Sun, and Zygouras. Joint work with Yu Gu (CMU).
+
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.
  
== November 19, 2020, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
+
== April 8, 2021, [http://www.math.ucsd.edu/~tiz161/ Tianyi Zheng] (UCSD) ==
 +
'''Random walks on wreath products and related groups'''
  
Title: '''TBA'''
+
Random walks on lamplighter groups were first considered by Kaimanovich and Vershik to provide examples of amenable groups with nontrivial Poisson boundary. Such processes can be understood rather explicitly, and provide guidance in the study of random walks on more complicated groups. In this talk we will discuss behavior of random walks on lamplighter groups, their extensions and some related groups which carry a similar semi-direct product structure.
  
Abstract: TBA
+
== April 15, 2021, [https://stat.wisc.edu/staff/levin-keith/ Keith Levin] (UW-Madison, Statistics)  ==
 +
'''Network Embeddings and Latent Space Models'''
  
== December 3, 2020, [https://www.math.wisc.edu/people/faculty-directory Tatyana Shcherbina] (UW-Madison) ==
+
Networks are data structures that describe relations among entities, such as friendships among people in a social network or synapses between neurons in a brain.  The field of statistical network analysis aims to develop network analogues of classical statistical techniques, and latent space models have emerged as the workhorse of this nascent field. Under these models, network formation is driven by unobserved geometric structure, in which each vertex in the network has an associated point in some metric space, called its latent position, that describes the (stochastic) behavior of the vertex in the network. In this talk, I will discuss some of my own work related to latent space models, focusing on 1) estimation of the vertex-level latent positions and 2) generating bootstrap replicates of network data.  Throughout the talk, I will make a point to highlight open problems and ongoing projects that are likely to be of interest to probabilists.
  
Title: '''TBA'''
+
== 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]  ==
 +
'''Modeling COVID-19 Spread in Universities'''
  
Abstract: TBA
+
University policy surrounding COVID-19 often involves big decisions informed by minimal data. Models are a tool to bridge this divide. I will describe some of the work that came out during Summer of 2020 to inform college reopening for Fall 2020. This includes a stochastic, agent-based model on a network for infection spread in residential colleges that I developed alongside a biologist, computer scientist, and group of students [https://arxiv.org/abs/2008.09597]. Time-permitting, I will describe a new project that aims to predict the impact of vaccination on infection spread in urban universities during the Fall 2021 semester. Disclaimer: I self-identify as a "pure" probabilist who typically proves theorems about particle systems [http://www.mathjunge.com/research]. These projects arose from my feeling compelled to help out to the best of my abilities during the height of the pandemic.
  
== December 10, 2020, [https://www.ewbates.com/ Erik Bates] (UW-Madison) ==
+
== April 22, 2021, [https://www.maths.ox.ac.uk/people/benjamin.fehrman Benjamin Fehrman] (Oxford) ==
 +
'''Non-equilibrium fluctuations in interacting particle systems and conservative stochastic PDE'''
  
Title: '''TBA'''
+
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.
  
Abstract: TBA
+
In this talk, which is based on joint work with Benjamin Gess, I will introduce a continuum model for simulating rare events in the zero range and symmetric simple exclusion process.  The model is based on an approximating sequence of stochastic partial differential equations with nonlinear, conservative noise.  The solutions capture to first-order the central limit fluctuations of the particle system, and they correctly simulate rare events in terms of a large deviations principle.
  
 +
== April 29, 2021, [http://www.stats.ox.ac.uk/~martin/ James Martin] (Oxford)  ==
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 17:31, 5 April 2021


Spring 2021

Thursdays in 901 Van Vleck Hall at 2:30 PM, unless otherwise noted. We usually end for questions at 3:20 PM.

IMPORTANT: In Spring 2021 the seminar is being run online. ZOOM LINK

If you would like to sign up for the email list to receive seminar announcements then please join our group.

January 28, 2021, no seminar

February 4, 2021, Hong-Bin Chen (Courant Institute, NYU)

Dynamic polymers: invariant measures and ordering by noise

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.

February 11, 2021, Kevin Yang (Stanford)

Non-stationary fluctuations for some non-integrable models

We will discuss recent progress on weak KPZ universality and non-integrable particle systems, including long-range models and slow bond models. The approach is based on a preliminary step in a non-stationary (first-order) Boltzmann-Gibbs principle. We will also discuss the full non-stationary Boltzmann-Gibbs principle itself and pieces of its proof.

February 18, 2021, Ilya Chevyrev (Edinburgh)

Signature moments to characterize laws of stochastic processes

The normalized sequence of moments characterizes the law of any finite-dimensional random variable. In this talk, I will describe an extension of this result to path-valued random variables, i.e. stochastic processes, by using the normalized sequence of signature moments. I will show how these moments define a metric for laws of stochastic processes. This metric can be efficiently estimated from finite samples, even if the stochastic processes themselves evolve in high-dimensional state spaces. As an application, I will describe a non-parametric two-sample hypothesis test for laws of stochastic processes.

February 25, 2021, Roger Van Peski (MIT)

Random matrices, random groups, singular values, and symmetric functions

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.

March 4, 2021, Roland Bauerschmidt (Cambridge)

The Coleman correspondence at the free fermion point

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. I will first review this correspondence in the explicit instance of the massless Gaussian free field and massless Euclidean Dirac fermions. I will then present a result that extends this correspondence to the non-Gaussian `massless' sine-Gordon field on $R^2$ at $\beta=4\pi$ and massive Dirac fermions. This is an instance of Coleman's prediction that the `massless' sine-Gordon model and the massive Thirring model are equivalent. We use this correspondence to show that correlations of the `massless' sine-Gordon model decay exponentially for $\beta=4\pi$. This is joint work with C. Webb (arXiv:2010.07096).

March 11, 2021, Sevak Mkrtchyan (Rochester)

The limit shape of the Leaky Abelian Sandpile Model

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.

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.

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.

March 18, 2021, Theo Assiotis (Edinburgh)

On the joint moments of characteristic polynomials of random unitary matrices

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

March 25, 2021, Wlodzimierz Bryc (Cincinnati)

Fluctuations of particle density for open ASEP

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

The talk is based on past and ongoing projects with Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.

April 1, 2021, Zoe Huang (Duke University)

Motion by mean curvature in interacting particle systems

There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et. al there were two nontrivial stationary distributions.

April 8, 2021, Tianyi Zheng (UCSD)

Random walks on wreath products and related groups

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.

April 15, 2021, Keith Levin (UW-Madison, Statistics)

Network Embeddings and Latent Space Models

Networks are data structures that describe relations among entities, such as friendships among people in a social network or synapses between neurons in a brain. The field of statistical network analysis aims to develop network analogues of classical statistical techniques, and latent space models have emerged as the workhorse of this nascent field. Under these models, network formation is driven by unobserved geometric structure, in which each vertex in the network has an associated point in some metric space, called its latent position, that describes the (stochastic) behavior of the vertex in the network. In this talk, I will discuss some of my own work related to latent space models, focusing on 1) estimation of the vertex-level latent positions and 2) generating bootstrap replicates of network data. Throughout the talk, I will make a point to highlight open problems and ongoing projects that are likely to be of interest to probabilists.

April 16, 2021, Matthew Junge (CUNY) FRIDAY at 2:25pm, joint with ACMS

Modeling COVID-19 Spread in Universities

University policy surrounding COVID-19 often involves big decisions informed by minimal data. Models are a tool to bridge this divide. I will describe some of the work that came out during Summer of 2020 to inform college reopening for Fall 2020. This includes a stochastic, agent-based model on a network for infection spread in residential colleges that I developed alongside a biologist, computer scientist, and group of students [1]. Time-permitting, I will describe a new project that aims to predict the impact of vaccination on infection spread in urban universities during the Fall 2021 semester. Disclaimer: I self-identify as a "pure" probabilist who typically proves theorems about particle systems [2]. These projects arose from my feeling compelled to help out to the best of my abilities during the height of the pandemic.

April 22, 2021, Benjamin Fehrman (Oxford)

Non-equilibrium fluctuations in interacting particle systems and conservative stochastic PDE

Abstract: Interacting particle systems have found diverse applications in mathematics and several related fields, including statistical physics, population dynamics, and machine learning. We will focus, in particular, on the zero range process and the symmetric simple exclusion process. The large-scale behavior of these systems is essentially deterministic, and is described in terms of a hydrodynamic limit. However, the particle process does exhibit large fluctuations away from its mean. Such deviations, though rare, can have significant consequences---such as a concentration of energy or the appearance of a vacuum---which make them important to understand and simulate.

In this talk, which is based on joint work with Benjamin Gess, I will introduce a continuum model for simulating rare events in the zero range and symmetric simple exclusion process. The model is based on an approximating sequence of stochastic partial differential equations with nonlinear, conservative noise. The solutions capture to first-order the central limit fluctuations of the particle system, and they correctly simulate rare events in terms of a large deviations principle.

April 29, 2021, James Martin (Oxford)

Past Seminars