Difference between revisions of "Probability Seminar"

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__NOTOC__
 
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[[Probability | Back to Probability Group]]
  
= Fall 2021 =
+
= Fall 2022 =
  
 
<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>  
<b>We usually end for questions at 3:20 PM.</b>
+
 
 +
We usually end for questions at 3:20 PM.
  
 
[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]
 
[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].
 
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 22, 2022, in person: [https://sites.google.com/site/pierreyvesgl/home Pierre Yves Gaudreau Lamarre] (University of Chicago)    ==
 +
 +
'''Moments of the Parabolic Anderson Model with Asymptotically Singular Noise'''
 
   
 
   
== January 28, 2021, no seminar ==
+
The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.
 +
   
 +
One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.
 +
 +
In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.
 +
 +
This talk is based on a joint work with Promit Ghosal and Yuchen Liao.
  
== February 4, 2021, [https://cims.nyu.edu/~hbchen/ Hong-Bin Chen] (Courant Institute, NYU) ==
+
== September 29, 2022, in person: Christian Gorski (Northwestern University)   ==
  
'''Dynamic polymers: invariant measures and ordering by noise'''
+
'''Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees'''
  
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.
+
I'll present strict monotonicity results for first passage percolation (FPP) on bounded degree graphs which either have strict polynomial growth (uniform upper and lower volume growth bounds of the same polynomial degree) or are quasi-isometric to a tree; the case of the standard Cayley graph of Z^d is due to van den Berg and Kesten (1993). Roughly speaking, if we use two different weight distributions to perform FPP on a fixed graph, and one of the distributions is "larger" than the other and "subcritical" in some appropriate sense, then the expected passage times with respect to that distribution exceed those of the other distribution by an amount proportional to the graph distance.  
 +
If "larger" here refers to stochastic domination of measures, this result is closely related to "absolute continuity with respect to the expected empirical measure," that is, the fact that long geodesics "use all possible weights". If "larger" here refers to variability (another ordering on measures), then a strict monotonicity theorem holds if and only if the graph also satisfies a condition we call "admitting detours". I intend to sketch the proof of absolute continuity, and, if time allows, give some indication of the difficulties that arise when proving strict monotonicity with respect to variability.
  
== February 11, 2021, [https://mathematics.stanford.edu/people/kevin-yang Kevin Yang] (Stanford) ==
+
== October 6, 2022, in person: [https://danielslonim.github.io/ Daniel Slonim] (University of Virginia)   ==  
  
'''Non-stationary fluctuations for some non-integrable models'''
+
'''Random Walks in (Dirichlet) Random Environments with Jumps on Z'''
  
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 introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random Walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.  
  
== February 18, 2021, [https://ilyachevyrev.wordpress.com Ilya Chevyrev] (Edinburgh) ==
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== October 13, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www.maths.univ-evry.fr/pages_perso/loukianova/ Dasha Loukianova] (Université d'Évry Val d'Essonne)   ==
  
'''Signature moments to characterize laws of stochastic processes'''
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'''"In law" ergodic theorem for the environment viewed from Sinaï's walk'''
  
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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For Sinaï's walk <math>\scriptsize(X_k)</math> we show that the empirical measure of the environment seen from the particle converges in law to some random measure. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov.  As a consequence an "in law" ergodic theorem holds for additive functionals of the environment's chain.  When the limit in this theorem is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov. In particular, we show an LLN and a mixed CLT for the sums <math>\scriptsize\sum_{k=1}^nf(\Delta X_k)</math> where <math>\scriptsize f</math> is bounded and
 +
depending on the steps <math>\scriptsize\Delta X_k:=X_{k+1}-X_k</math>.
  
== February 25, 2021, [https://math.mit.edu/directory/profile.php?pid=2121 Roger Van Peski] (MIT) ==
+
== October 20, 2022, '''4pm, VV911''', in person: [https://tavarelab.cancerdynamics.columbia.edu/ Simon Tavaré] (Columbia University) ==
 +
''Note the unusual time and room!''
  
'''Random matrices, random groups, singular values, and symmetric functions'''
+
'''An introduction to counts-of-counts data'''
  
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.
+
Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers ''C<sub>1</sub>, C<sub>2</sub>, …'' of species represented once, twice, in a sample of size
  
== March 4, 2021,  [http://www.statslab.cam.ac.uk/~rb812/ Roland Bauerschmidt] (Cambridge) ==
+
''N = C<sub>1</sub> + 2 C<sub>2</sub> + 3 C<sub>3</sub> + <sup>….</sup>''  containing ''S = C<sub>1</sub> + C<sub>2</sub> + <sup>…</sup>'' species; the vector ''C ='' ''(C<sub>1</sub>, C<sub>2</sub>, …)'' gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.
  
'''The Coleman correspondence at the free fermion point'''
+
In this talk I will outline some of the stochastic models used to model the distribution of ''C,'' and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of ''S'' in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.
  
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, [https://people.math.rochester.edu/faculty/smkrtchy/ Sevak Mkrtchyan] (Rochester)  ==
+
''References''
  
'''The limit shape of the Leaky Abelian Sandpile Model'''
+
[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943
  
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.
+
[2] Arratia R, Barbour AD & Tavaré S. ''Logarithmic Combinatorial Structures,'' EMS, 2002
  
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.
+
[3] Ewens WJ. Theoret Popul Biol, 3, 1972
  
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.
+
[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)
  
== March 18, 2021, [https://sites.google.com/view/theoassiotis/home Theo Assiotis] (Edinburgh) ==
+
== October 27, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://www-users.cse.umn.edu/~arnab/ Arnab Sen] (University of Minnesota, Twin Cities) ==  
  
'''On the joint moments of characteristic polynomials of random unitary matrices'''
+
'''Maximum weight matching on sparse graphs'''
 
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, [https://homepages.uc.edu/~brycwz/ Wlodzimierz Bryc] (Cincinnati) ==
+
We consider the maximum weight matching of a finite bounded degree graph whose edges have i.i.d. random weights. It is natural to ask whether the weight of the maximum weight matching follows a central limit theorem. We obtain an affirmative answer to the above question in the case when the weight distribution is exponential and the graphs are locally tree-like. The key component of the proof involves a cavity analysis on arbitrary bounded degree trees which yields a correlation decay for the maximum weight matching. The central limit theorem holds if we take the underlying graph to be also random with i.i.d. degree distribution (configuration model).
'''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.
+
This is joint work with Wai-Kit Lam.
  
The talk is based on past and ongoing projects with  Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.
+
== November 3, 2022, in person: [https://www.ias.edu/scholars/sky-yang-cao Sky Cao] (Institute for Advanced Study)  ==
  
== April 1, 2021, [https://sites.google.com/view/xiangying-huangs-home-page/home Zoe Huang] (Duke University)  ==
+
'''Exponential decay of correlations in finite gauge group lattice gauge theories'''
'''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.
+
Lattice gauge theories with finite gauge groups are statistical mechanical models, very much akin to the Ising model, but with some twists. In this talk, I will describe how to show exponential decay of correlations for these models at low temperatures. This is based on joint work with Arka Adhikari.
  
== April 8, 2021, [http://www.math.ucsd.edu/~tiz161/ Tianyi Zheng] (UCSD) ==
+
== November 10, 2022, in person: [https://ifds.info/david-clancy/ David Clancy] (UW-Madison)   ==  
'''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.
+
'''Component Sizes of the degree corrected stochastic blockmodel'''
  
== April 15, 2021, [https://stat.wisc.edu/staff/levin-keith/ Keith Levin] (UW-Madison, Statistics)  ==
+
The stochastic blockmodel (SBM) is a simple probabilistic model for graphs which exhibit clustering and is used to test algorithms for detecting these clusters. Each vertex is assigned a type ''i = 1, 2, ..., m'' and edges are included independently with probability depending on the types of the two incident vertices. The degree corrected SBM (DCSBM) exhibits similar clustering behavior but allows for inhomogeneous degree distributions. The sizes of connected components for these graph models are not well understood unless ''m = 1'' or the SBM is a random bipartite graph. We show that under fairly general conditions, the asymptotic sizes of connected components in the DCSBM can be precisely described in terms of a multiparameter and multidimensional random field. Not only that, but we describe the asymptotic proportion of vertices of each type in each of the macroscopic connected components. This talk is based on joint work with Vitalii Konarovskyi and Vlada Limic.
'''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.
+
== November 17, 2022, [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM]: [https://sites.google.com/site/leandroprpimentel/ Leandro Pimentel] (Federal University of Rio de Janeiro)   ==
  
== 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]  ==
+
'''Integration by Parts and the KPZ Two-Point Function'''
'''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 [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.
+
In this talk we will consider two models within Kardar-Parisi-Zhang (KPZ) universality class, and apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point correlation function, the polymer end-point distribution and the second derivative of the variance of the associated height function. Besides that, we will further develop an adaptation of Malliavin-Stein method that quantifies asymptotic independence with respect to the initial data.  
  
== April 22, 2021, [https://www.maths.ox.ac.uk/people/benjamin.fehrman Benjamin Fehrman] (Oxford) ==
+
== December 1, 2022, in person: [https://cims.nyu.edu/~ajd594/ Alex Dunlap] (Courant Institute)   ==  
'''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.
+
'''The nonlinear stochastic heat equation in the critical dimension'''
  
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.
+
I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on current joint work with Cole Graham and older joint work with Yu Gu.
  
== April 29, 2021, [http://www.stats.ox.ac.uk/~martin/ James Martin] (Oxford) ==
+
== December 8, 2022, in person: [https://sites.northwestern.edu/juliagaudio/ Julia Gaudio] (Northwestern University)   ==  
'''The environment seen from a geodesic in last-passage percolation, and the TASEP seen from a second-class particle'''
 
  
We study directed last-passage percolation in $\mathbb{Z}^2$ with i.i.d.
+
'''Finding Communities in Networks'''
exponential weights. What does a geodesic path look like locally, and
+
how do the weights on and nearby the geodesic behave? We show
+
Networks are used to represent physical, biological, and social systems. Many networks exhibit community structure, meaning that there are two or more groups of nodes which are densely connected. Identifying these communities gives valuable insights about the latent features of the nodes. Community detection has been used in a wide array of applications including online advertising, recommender systems (e.g., Netflix), webpage sorting, fraud detection, and neurobiology.
convergence of the distribution of the "environment" as seen from a
+
typical point along the geodesic in a given direction, as its length
+
I will present my work on efficient algorithms for community detection in three contexts. <br>
goes to infinity. We describe the limiting distribution, and can
+
(1) Censored networks: How can we identify communities when some connectivity information is missing? <br>
calculate various quantities such as the density function of a typical
+
(2) Higher-order networks: Beyond pairwise relationships <br>
weight, or the proportion of "corners" along the path. The analysis
+
(3) Multiple correlated networks: How can we effectively combine data from multiple networks? <br>
involves a link with the TASEP (totally asymmetric simple exclusion
+
process) seen from an isolated second-class particle, and we obtain
+
Joint work with: Souvik Dhara, Nirmit Joshi, Elchanan Mossel, Miklós Rácz, Colin Sandon, and Anirudh Sridhar
some new convergence and ergodicity results for that process. The talk
 
is based on joint work with Allan Sly and Lingfu Zhang.
 
  
  
 
[[Past Seminars]]
 
[[Past Seminars]]

Latest revision as of 23:32, 27 November 2022

Back to Probability Group

Fall 2022

Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom

We usually end for questions at 3:20 PM.

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 our group.


September 22, 2022, in person: Pierre Yves Gaudreau Lamarre (University of Chicago)

Moments of the Parabolic Anderson Model with Asymptotically Singular Noise

The Parabolic Anderson Model (PAM) is a stochastic partial differential equation that describes the time-evolution of particle system with the following dynamics: Each particle in the system undergoes a diffusion in space, and as they are moving through space, the particles can either multiply or get killed at a rate that depends on a random environment.

One of the fundamental problems in the theory of the PAM is to understand its behavior at large times. More specifically, the solution of the PAM at large times tends to be intermittent, meaning that most of the particles concentrate in small regions where the environment is most favorable for particle multiplication.

In this talk, we discuss a new technique to study intermittency in the PAM with a singular random environment. In short, the technique consists of approximating the singular PAM with a regularized version that becomes increasingly singular as time goes to infinity.

This talk is based on a joint work with Promit Ghosal and Yuchen Liao.

September 29, 2022, in person: Christian Gorski (Northwestern University)

Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

I'll present strict monotonicity results for first passage percolation (FPP) on bounded degree graphs which either have strict polynomial growth (uniform upper and lower volume growth bounds of the same polynomial degree) or are quasi-isometric to a tree; the case of the standard Cayley graph of Z^d is due to van den Berg and Kesten (1993). Roughly speaking, if we use two different weight distributions to perform FPP on a fixed graph, and one of the distributions is "larger" than the other and "subcritical" in some appropriate sense, then the expected passage times with respect to that distribution exceed those of the other distribution by an amount proportional to the graph distance. If "larger" here refers to stochastic domination of measures, this result is closely related to "absolute continuity with respect to the expected empirical measure," that is, the fact that long geodesics "use all possible weights". If "larger" here refers to variability (another ordering on measures), then a strict monotonicity theorem holds if and only if the graph also satisfies a condition we call "admitting detours". I intend to sketch the proof of absolute continuity, and, if time allows, give some indication of the difficulties that arise when proving strict monotonicity with respect to variability.

October 6, 2022, in person: Daniel Slonim (University of Virginia)

Random Walks in (Dirichlet) Random Environments with Jumps on Z

We introduce the model of random walks in random environments (RWRE), which are random Markov chains on the integer lattice. These random walks are well understood in the nearest-neighbor, one-dimensional case due to reversibility of almost every Markov chain. For example, directional transience and limiting speed can be characterized in terms of simple expectations involving the transition probabilities at a single site. The reversibility is lost, however, if we go up to higher dimensions or relax the nearest-neighbor assumption by allowing jumps, and therefore much less is known in these models. Despite this non-reversibility, certain special cases have proven to be more tractable. Random Walks in Dirichlet environments (RWDE), where the transition probability vectors are drawn according to a Dirichlet distribution, have been fruitfully studied in the nearest-neighbor, higher dimensional setting. We look at RWDE in one dimension with jumps and characterize when the walk is ballistic: that is, when it has non-zero limiting velocity. It turns out that in this model, there are two factors which can cause a directionally transient walk to have zero limiting speed: finite trapping and large-scale backtracking. Finite trapping involves finite subsets of the graph where the walk is liable to get trapped for a long time. It is a highly local phenomenon that depends heavily on the structure of the underlying graph. Large-scale backtracking is a more global and one-dimensional phenomenon. The two operate "independently" in the sense that either can occur with or without the other. Moreover, if neither factor on its own is enough to cause zero speed, then the walk is ballistic, so the two factors cannot conspire together to slow a walk down to zero speed if neither is sufficient to do so on its own. This appearance of two independent factors affecting ballisticity is a new feature not seen in any previously studied RWRE models.

October 13, 2022, ZOOM: Dasha Loukianova (Université d'Évry Val d'Essonne)

"In law" ergodic theorem for the environment viewed from Sinaï's walk

For Sinaï's walk [math]\displaystyle{ \scriptsize(X_k) }[/math] we show that the empirical measure of the environment seen from the particle converges in law to some random measure. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov. As a consequence an "in law" ergodic theorem holds for additive functionals of the environment's chain. When the limit in this theorem is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov. In particular, we show an LLN and a mixed CLT for the sums [math]\displaystyle{ \scriptsize\sum_{k=1}^nf(\Delta X_k) }[/math] where [math]\displaystyle{ \scriptsize f }[/math] is bounded and depending on the steps [math]\displaystyle{ \scriptsize\Delta X_k:=X_{k+1}-X_k }[/math].

October 20, 2022, 4pm, VV911, in person: Simon Tavaré (Columbia University)

Note the unusual time and room!

An introduction to counts-of-counts data

Counts-of-counts data arise in many areas of biology and medicine, and have been studied by statisticians since the 1940s. One of the first examples, discussed by R. A. Fisher and collaborators in 1943 [1], concerns estimation of the number of unobserved species based on summary counts of the number of species observed once, twice, … in a sample of specimens. The data are summarized by the numbers C1, C2, … of species represented once, twice, … in a sample of size

N = C1 + 2 C2 + 3 C3 + ….  containing S = C1 + C2 + species; the vector C = (C1, C2, …) gives the counts-of-counts. Other examples include the frequencies of the distinct alleles in a human genetics sample, the counts of distinct variants of the SARS-CoV-2 S protein obtained from consensus sequencing experiments, counts of sizes of components in certain combinatorial structures [2], and counts of the numbers of SNVs arising in one cell, two cells, … in a cancer sequencing experiment.

In this talk I will outline some of the stochastic models used to model the distribution of C, and some of the inferential issues that come from estimating the parameters of these models. I will touch on the celebrated Ewens Sampling Formula [3] and Fisher’s multiple sampling problem concerning the variance expected between values of S in samples taken from the same population [3]. Variants of birth-death-immigration processes can be used, for example when different variants grow at different rates. Some of these models are mechanistic in spirit, others more statistical. For example, a non-mechanistic model is useful for describing the arrival of covid sequences at a database. Sequences arrive one at a time, and are either a new variant, or a copy of a variant that has appeared before. The classical Yule process with immigration provides a starting point to model this process, as I will illustrate.


References

[1] Fisher RA, Corbet AS & Williams CB. J Animal Ecology, 12, 1943

[2] Arratia R, Barbour AD & Tavaré S. Logarithmic Combinatorial Structures, EMS, 2002

[3] Ewens WJ. Theoret Popul Biol, 3, 1972

[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)

October 27, 2022, ZOOM: Arnab Sen (University of Minnesota, Twin Cities)

Maximum weight matching on sparse graphs

We consider the maximum weight matching of a finite bounded degree graph whose edges have i.i.d. random weights. It is natural to ask whether the weight of the maximum weight matching follows a central limit theorem. We obtain an affirmative answer to the above question in the case when the weight distribution is exponential and the graphs are locally tree-like. The key component of the proof involves a cavity analysis on arbitrary bounded degree trees which yields a correlation decay for the maximum weight matching. The central limit theorem holds if we take the underlying graph to be also random with i.i.d. degree distribution (configuration model).

This is joint work with Wai-Kit Lam.

November 3, 2022, in person: Sky Cao (Institute for Advanced Study)

Exponential decay of correlations in finite gauge group lattice gauge theories

Lattice gauge theories with finite gauge groups are statistical mechanical models, very much akin to the Ising model, but with some twists. In this talk, I will describe how to show exponential decay of correlations for these models at low temperatures. This is based on joint work with Arka Adhikari.

November 10, 2022, in person: David Clancy (UW-Madison)

Component Sizes of the degree corrected stochastic blockmodel

The stochastic blockmodel (SBM) is a simple probabilistic model for graphs which exhibit clustering and is used to test algorithms for detecting these clusters. Each vertex is assigned a type i = 1, 2, ..., m and edges are included independently with probability depending on the types of the two incident vertices. The degree corrected SBM (DCSBM) exhibits similar clustering behavior but allows for inhomogeneous degree distributions. The sizes of connected components for these graph models are not well understood unless m = 1 or the SBM is a random bipartite graph. We show that under fairly general conditions, the asymptotic sizes of connected components in the DCSBM can be precisely described in terms of a multiparameter and multidimensional random field. Not only that, but we describe the asymptotic proportion of vertices of each type in each of the macroscopic connected components. This talk is based on joint work with Vitalii Konarovskyi and Vlada Limic.

November 17, 2022, ZOOM: Leandro Pimentel (Federal University of Rio de Janeiro)

Integration by Parts and the KPZ Two-Point Function

In this talk we will consider two models within Kardar-Parisi-Zhang (KPZ) universality class, and apply the integration by parts formula from Malliavin calculus to establish a key relation between the two-point correlation function, the polymer end-point distribution and the second derivative of the variance of the associated height function. Besides that, we will further develop an adaptation of Malliavin-Stein method that quantifies asymptotic independence with respect to the initial data.

December 1, 2022, in person: Alex Dunlap (Courant Institute)

The nonlinear stochastic heat equation in the critical dimension

I will discuss a two-dimensional stochastic heat equation with a nonlinear noise strength, and consider a limit in which the correlation length of the noise is taken to 0 but the noise is attenuated by a logarithmic factor. The limiting pointwise statistics can be related to a stochastic differential equation in which the diffusivity solves a PDE somewhat reminiscent of the porous medium equation. This relationship is established through the theory of forward-backward SDEs. I will also explain several cases in which the PDE can be solved explicitly, some of which correspond to known probabilistic models. This talk will be based on current joint work with Cole Graham and older joint work with Yu Gu.

December 8, 2022, in person: Julia Gaudio (Northwestern University)

Finding Communities in Networks

Networks are used to represent physical, biological, and social systems. Many networks exhibit community structure, meaning that there are two or more groups of nodes which are densely connected. Identifying these communities gives valuable insights about the latent features of the nodes. Community detection has been used in a wide array of applications including online advertising, recommender systems (e.g., Netflix), webpage sorting, fraud detection, and neurobiology.

I will present my work on efficient algorithms for community detection in three contexts.
(1) Censored networks: How can we identify communities when some connectivity information is missing?
(2) Higher-order networks: Beyond pairwise relationships
(3) Multiple correlated networks: How can we effectively combine data from multiple networks?

Joint work with: Souvik Dhara, Nirmit Joshi, Elchanan Mossel, Miklós Rácz, Colin Sandon, and Anirudh Sridhar


Past Seminars