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[[Probability | Back to Probability Group]]


= Spring 2021 =
* '''When''': Thursdays at 2:30 pm
* '''Where''': 901 Van Vleck Hall
* '''Organizers''': Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
* '''To join the probability seminar mailing list:''' email probsem+subscribe@g-groups.wisc.edu.
* '''To subscribe seminar lunch announcements:''' email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu


<b>Thursdays 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>
 
 
= Fall 2024 =
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>


<b> IMPORTANT: </b> In Spring 2021 the seminar is being run online. [https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK]
We usually end for questions at 3:20 PM.


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 5, 2024: ==
No seminar
== January 28, 2021, no seminar  ==


== February 4, 2021, [https://cims.nyu.edu/~hbchen/ Hong-Bin Chen] (Courant Institute, NYU) ==
== September 12, 2024: Hongchang Ji (UW-Madison) ==
'''Spectral edge of non-Hermitian random matrices'''


'''Dynamic polymers: invariant measures and ordering by noise'''
We report recent progress on spectra of so-called deformed i.i.d. matrices. They are square non-Hermitian random matrices of the form $A+X$ where $X$ has centered i.i.d. entries and $A$ is a deterministic bias, and $A$ and $X$ are on the same scale so that their contributions to the spectrum of $A+X$ are comparable. Under this setting, we present two recent results concerning universal patterns arising in eigenvalue statistics of $A+X$ around its boundary, on macroscopic and microscopic scales. The first result shows that the macroscopic eigenvalue density of $A+X$ typically has a jump discontinuity around the boundary of its support, which is a distinctive feature of $X$ by the \emph{circular law}. The second result is edge universality for deformed non-Hermitian matrices; it shows that the local eigenvalue statistics of $A+X$ around a typical (jump) boundary point is universal, i.e., matches with those of a Ginibre matrix $X$ with i.i.d. standard Gaussian entries.


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.
Based on joint works with A. Campbell, G. Cipolloni, and L. Erd\H{o}s.


== February 11, 2021, [https://mathematics.stanford.edu/people/kevin-yang Kevin Yang] (Stanford)  ==


'''Non-stationary fluctuations for some non-integrable models'''
== September 19, 2024: Miklos Racz (Northwestern) ==
'''The largest common subtree of uniform attachment trees'''


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.
Consider two independent uniform attachment trees with n nodes each -- how large is their largest common subtree? Our main result gives a lower bound of n^{0.83}. We also give some upper bounds and bounds for general random tree growth models. This is based on joint work with Johannes Bäumler, Bas Lodewijks, James Martin, Emil Powierski, and Anirudh Sridhar.


== February 18, 2021, [https://ilyachevyrev.wordpress.com Ilya Chevyrev] (Edinburgh) ==
== September 26, 2024: Dmitry Krachun (Princeton) ==
'''A glimpse of universality in critical planar lattice models'''


'''Signature moments to characterize laws of stochastic processes'''
Abstract: Many models of statistical mechanics are defined on a lattice, yet they describe behaviour of objects in our seemingly isotropic world. It is then natural to ask why, in the small mesh size limit, the directions of the lattice disappear. Physicists' answer to this question is partially given by the Universality hypothesis, which roughly speaking states that critical properties of a physical system do not depend on the lattice or fine properties of short-range interactions but only depend on the spatial dimension and the symmetry of the possible spins. Justifying the reasoning behind the universality hypothesis mathematically seems virtually impossible and so other ideas are needed for a rigorous derivation of universality even in the simplest of setups.


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.
In this talk I will explain some ideas behind the recent result which proves rotational invariance of the FK-percolation model. In doing so, we will see how rotational invariance is related to universality among a certain one-dimensional family of planar lattices and how the latter can be proved using exact integrability of the six-vertex model using Bethe ansatz.  


== February 25, 2021, [https://math.mit.edu/directory/profile.php?pid=2121 Roger Van Peski] (MIT)  ==
Based on joint works with Hugo Duminil-Copin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, and Tatiana Tikhonovskaia.


'''Random matrices, random groups, singular values, and symmetric functions'''
== October 3, 2024: Joshua Cape (UW-Madison) ==
'''A new random matrix: motivation, properties, and applications'''


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.
In this talk, we introduce and study a new random matrix whose entries are dependent and discrete valued. This random matrix is motivated by problems in multivariate analysis and nonparametric statistics. We establish its asymptotic properties and provide comparisons to existing results for independent entry random matrix models. We then apply our results to two problems: (i) community detection, and (ii) principal submatrix localization. Based on joint work with Jonquil Z. Liao.


== March 4, 2021,  [http://www.statslab.cam.ac.uk/~rb812/ Roland Bauerschmidt] (Cambridge) ==
== October 10, 2024: Midwest Probability Colloquium ==
N/A


'''The Coleman correspondence at the free fermion point'''
== October 17, 2024: Kihoon Seong (Cornell) ==
'''Gaussian fluctuations of focusing Φ^4 measure around the soliton manifold'''


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 explain the central limit theorem for the focusing Φ^4 measure in the infinite volume limit. The focusing Φ^4 measure, an invariant Gibbs measure for the nonlinear Schrödinger equation, was first studied by Lebowitz, Rose, and Speer (1988), and later extended by Bourgain (1994), Brydges and Slade (1996), and Carlen, Fröhlich, and Lebowitz (2016).  
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)  ==
Rider previously showed that this measure is strongly concentrated around a family of minimizers of the associated Hamiltonian, known as the soliton manifold. In this talk, I will discuss the fluctuations around this soliton manifold. Specifically, we show that the scaled field under the focusing Φ^4 measure converges to white noise in the infinite volume limit, thus identifying the next-order fluctuations, as predicted by Rider.


'''The limit shape of the Leaky Abelian Sandpile Model'''
This talk is based on joint work with 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.
== October 24, 2024: Jacob Richey (Alfred Renyi Institute) ==
'''Stochastic abelian particle systems and self-organized criticality'''


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.
Abstract: Activated random walk (ARW) is an 'abelian' particle system that conjecturally exhibits complex behaviors which were first described by physicists in the 1990s, namely self organized criticality and hyperuniformity. I will discuss recent results for ARW and the stochastic sandpile (a related model) on Z and other graphs, plus many open questions.


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.
== October 31, 2024: David Clancy (UW-Madison) ==
'''Likelihood landscape on a known phylogeny'''


== March 18, 2021, [https://sites.google.com/view/theoassiotis/home Theo Assiotis] (Edinburgh) ==
Abstract: Over time, ancestral populations evolve to become separate species. We can represent this history as a tree with edge lengths where the leaves are the modern-day species. If we know the precise topology of the tree (i.e. the precise evolutionary relationship between all the species), then we can imagine traits (their presence or absence) being passed down according to a symmetric 2-state continuous-time Markov chain. The branch length becomes the probability a parent species has a trait while the child species does not. This length is unknown, but researchers have observed they can get pretty good estimates using maximum likelihood estimation and only the leaf data despite the fact that the number of critical points for the log-likelihood grows exponentially fast in the size of the tree. In this talk, I will discuss why this MLE approach works by showing that the population log-likelihood is strictly concave and smooth in a neighborhood around the true branch length parameters and the size.


'''On the joint moments of characteristic polynomials of random unitary matrices'''
This talk is based on joint work with Hanbaek Lyu, Sebastien Roch and Allan Sly.
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) ==
== November 7, 2024: Zoe Huang (UNC Chapel Hill) ==
'''Fluctuations of particle density  for open ASEP'''
'''Cutoff for Cayley graphs of nilpotent groups'''


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.
Abstract: Abstract:  We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes $G=G(n)$, whose ranks and nilpotency classes are uniformly bounded. For some $k=k(n)$ such that $1\ll\log k \ll \log |G|$, we pick a random set of generators $S=S(n)$ by sampling $k$ elements $Z_1,\ldots,Z_k$ from $G$ uniformly at random with replacement, and set $S:=\{Z_j^{\pm 1}:1 \le j\le k \}$. We show that the simple random walk on Cay$(G,S)$ exhibits cutoff with high probability. Some of our results apply to a general set of generators. Namely, we show that there is a constant $c>0$, depending only on the rank and the nilpotency class of $G$, such that for all symmetric sets of generators $S$ of size at most $ \frac{c\log |G|}{\log \log |G|}$, the spectral gap and the $\varepsilon$-mixing time of the simple random walk $X=(X_t)_{t\geq 0}$ on Cay$(G,S)$ are asymptotically the same as those of the projection of $X$ to the abelianization of $G$, given by $[G,G]X_t$. In particular, $X$ exhibits cutoff if and only if its projection does. Based on joint work with Jonathan Hermon.


The talk is based on past and ongoing projects with  Alexey Kuznetzov, Yizao Wang and Jacek Wesolowski.
== November 14, 2024: Nabarun Deb (University of Chicago) ==
Mean-Field fluctuations in Ising models and posterior prediction intervals in low signal-to-noise ratio regimes


== April 1, 2021, [https://sites.google.com/view/xiangying-huangs-home-page/home Zoe Huang] (Duke University) ==
Ising models have become central in probability, statistics, and machine learning. They naturally appear in the posterior distribution of regression coefficients under the linear model $Y = X\beta + \epsilon$, where $\epsilon \sim N(0, \sigma^2 I_n)$. This talk explores fluctuations of specific linear statistics under the Ising model, with a focus on applications in Bayesian linear regression.
'''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.
In the first part, we examine Ising models on "dense regular" graphs and characterize the limiting distribution of average magnetization across various temperature and magnetization regimes, extending previous results beyond the Curie-Weiss (complete graph) case. In the second part, we analyze posterior prediction intervals for linear statistics in low signal-to-noise ratio (SNR) scenarios, also known as the contiguity regime. Here, unlike standard Bernstein-von Mises results, the limiting distributions are highly sensitive to the choice of prior. We illustrate this dependency by presenting limiting laws under both correctly specified and misspecified priors.


This talk is based on joint work with Sumit Mukherjee and Seunghyun Li.


== April 8, 2021, [http://www.math.ucsd.edu/~tiz161/ Tianyi Zheng] (UCSD) ==
== November 21, 2024: Reza Gheissari (Northwestern) ==
'''Wetting and pre-wetting in (2+1)D solid-on-solid interfaces'''


== April 15, 2021, [https://stat.wisc.edu/staff/levin-keith/ Keith Levin] (UW-Madison, Statistics) ==
The (d+1)D-solid-on-solid model is a simple model of integer-valued height functions that approximates the low-temperature interface of an Ising model. When $d\ge 2$, with zero-boundary conditions, at low temperatures the surface is localized about height $0$, but when constrained to take only non-negative values entropic repulsion pushes it to take typical heights of $O(\log n)$.  I will describe the mechanism of entropic repulsion, and present results on how the picture changes when one introduces a competing force trying to keep the interface localized (either an external field or a reward for points where the height is exactly zero). Along the way, I will outline rich predictions for the shapes of level curves, and for metastability phenomena in the Glauber dynamics. Based on joint work with Eyal Lubetzky and Joseph Chen.


== 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]  ==
== November 28, 2024: Thanksgiving ==
No seminar


== April 22, 2021, TBA  ==
== December 5, 2024: Erik Bates (NC State) ==


== April 29, 2021, [http://www.stats.ox.ac.uk/~martin/ James Martin] (Oxford)  ==
'''Parisi formulas in multi-species and vector spin glass models'''


[[Past Seminars]]
The expression "Parisi formula" refers to a variational formula postulated by Parisi in 1980 to give the limiting free energy of the Sherrington--Kirkpatrick (SK) spin glass.  The SK model was originally conceived as a mean-field description for disordered magnetism, and has since become a mathematical prototype for frustrated disordered systems and high-complexity functions.  In recent years, there has been an effort to extend the Parisi framework to various generalizations of the SK model, raising new physical questions met with fresh mathematical challenges.  In this talk, I will share some developments in this evolving story.  Based on joint works with Leila Sloman and Youngtak Sohn.

Latest revision as of 20:39, 22 November 2024

Back to Probability Group

  • When: Thursdays at 2:30 pm
  • Where: 901 Van Vleck Hall
  • Organizers: Hanbaek Lyu, Tatyana Shcherbyna, David Clancy
  • To join the probability seminar mailing list: email probsem+subscribe@g-groups.wisc.edu.
  • To subscribe seminar lunch announcements: email lunchwithprobsemspeaker+subscribe@g-groups.wisc.edu

Past Seminars


Fall 2024

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

We usually end for questions at 3:20 PM.

September 5, 2024:

No seminar

September 12, 2024: Hongchang Ji (UW-Madison)

Spectral edge of non-Hermitian random matrices

We report recent progress on spectra of so-called deformed i.i.d. matrices. They are square non-Hermitian random matrices of the form $A+X$ where $X$ has centered i.i.d. entries and $A$ is a deterministic bias, and $A$ and $X$ are on the same scale so that their contributions to the spectrum of $A+X$ are comparable. Under this setting, we present two recent results concerning universal patterns arising in eigenvalue statistics of $A+X$ around its boundary, on macroscopic and microscopic scales. The first result shows that the macroscopic eigenvalue density of $A+X$ typically has a jump discontinuity around the boundary of its support, which is a distinctive feature of $X$ by the \emph{circular law}. The second result is edge universality for deformed non-Hermitian matrices; it shows that the local eigenvalue statistics of $A+X$ around a typical (jump) boundary point is universal, i.e., matches with those of a Ginibre matrix $X$ with i.i.d. standard Gaussian entries.

Based on joint works with A. Campbell, G. Cipolloni, and L. Erd\H{o}s.


September 19, 2024: Miklos Racz (Northwestern)

The largest common subtree of uniform attachment trees

Consider two independent uniform attachment trees with n nodes each -- how large is their largest common subtree? Our main result gives a lower bound of n^{0.83}. We also give some upper bounds and bounds for general random tree growth models. This is based on joint work with Johannes Bäumler, Bas Lodewijks, James Martin, Emil Powierski, and Anirudh Sridhar.

September 26, 2024: Dmitry Krachun (Princeton)

A glimpse of universality in critical planar lattice models

Abstract: Many models of statistical mechanics are defined on a lattice, yet they describe behaviour of objects in our seemingly isotropic world. It is then natural to ask why, in the small mesh size limit, the directions of the lattice disappear. Physicists' answer to this question is partially given by the Universality hypothesis, which roughly speaking states that critical properties of a physical system do not depend on the lattice or fine properties of short-range interactions but only depend on the spatial dimension and the symmetry of the possible spins. Justifying the reasoning behind the universality hypothesis mathematically seems virtually impossible and so other ideas are needed for a rigorous derivation of universality even in the simplest of setups.

In this talk I will explain some ideas behind the recent result which proves rotational invariance of the FK-percolation model. In doing so, we will see how rotational invariance is related to universality among a certain one-dimensional family of planar lattices and how the latter can be proved using exact integrability of the six-vertex model using Bethe ansatz.

Based on joint works with Hugo Duminil-Copin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, and Tatiana Tikhonovskaia.

October 3, 2024: Joshua Cape (UW-Madison)

A new random matrix: motivation, properties, and applications

In this talk, we introduce and study a new random matrix whose entries are dependent and discrete valued. This random matrix is motivated by problems in multivariate analysis and nonparametric statistics. We establish its asymptotic properties and provide comparisons to existing results for independent entry random matrix models. We then apply our results to two problems: (i) community detection, and (ii) principal submatrix localization. Based on joint work with Jonquil Z. Liao.

October 10, 2024: Midwest Probability Colloquium

N/A

October 17, 2024: Kihoon Seong (Cornell)

Gaussian fluctuations of focusing Φ^4 measure around the soliton manifold

I will explain the central limit theorem for the focusing Φ^4 measure in the infinite volume limit. The focusing Φ^4 measure, an invariant Gibbs measure for the nonlinear Schrödinger equation, was first studied by Lebowitz, Rose, and Speer (1988), and later extended by Bourgain (1994), Brydges and Slade (1996), and Carlen, Fröhlich, and Lebowitz (2016).

Rider previously showed that this measure is strongly concentrated around a family of minimizers of the associated Hamiltonian, known as the soliton manifold. In this talk, I will discuss the fluctuations around this soliton manifold. Specifically, we show that the scaled field under the focusing Φ^4 measure converges to white noise in the infinite volume limit, thus identifying the next-order fluctuations, as predicted by Rider.

This talk is based on joint work with Philippe Sosoe (Cornell).

October 24, 2024: Jacob Richey (Alfred Renyi Institute)

Stochastic abelian particle systems and self-organized criticality

Abstract: Activated random walk (ARW) is an 'abelian' particle system that conjecturally exhibits complex behaviors which were first described by physicists in the 1990s, namely self organized criticality and hyperuniformity. I will discuss recent results for ARW and the stochastic sandpile (a related model) on Z and other graphs, plus many open questions.

October 31, 2024: David Clancy (UW-Madison)

Likelihood landscape on a known phylogeny

Abstract: Over time, ancestral populations evolve to become separate species. We can represent this history as a tree with edge lengths where the leaves are the modern-day species. If we know the precise topology of the tree (i.e. the precise evolutionary relationship between all the species), then we can imagine traits (their presence or absence) being passed down according to a symmetric 2-state continuous-time Markov chain. The branch length becomes the probability a parent species has a trait while the child species does not. This length is unknown, but researchers have observed they can get pretty good estimates using maximum likelihood estimation and only the leaf data despite the fact that the number of critical points for the log-likelihood grows exponentially fast in the size of the tree. In this talk, I will discuss why this MLE approach works by showing that the population log-likelihood is strictly concave and smooth in a neighborhood around the true branch length parameters and the size.

This talk is based on joint work with Hanbaek Lyu, Sebastien Roch and Allan Sly.

November 7, 2024: Zoe Huang (UNC Chapel Hill)

Cutoff for Cayley graphs of nilpotent groups

Abstract: Abstract:  We consider the random Cayley graphs of a sequence of finite nilpotent groups of diverging sizes $G=G(n)$, whose ranks and nilpotency classes are uniformly bounded. For some $k=k(n)$ such that $1\ll\log k \ll \log |G|$, we pick a random set of generators $S=S(n)$ by sampling $k$ elements $Z_1,\ldots,Z_k$ from $G$ uniformly at random with replacement, and set $S:=\{Z_j^{\pm 1}:1 \le j\le k \}$. We show that the simple random walk on Cay$(G,S)$ exhibits cutoff with high probability. Some of our results apply to a general set of generators. Namely, we show that there is a constant $c>0$, depending only on the rank and the nilpotency class of $G$, such that for all symmetric sets of generators $S$ of size at most $ \frac{c\log |G|}{\log \log |G|}$, the spectral gap and the $\varepsilon$-mixing time of the simple random walk $X=(X_t)_{t\geq 0}$ on Cay$(G,S)$ are asymptotically the same as those of the projection of $X$ to the abelianization of $G$, given by $[G,G]X_t$. In particular, $X$ exhibits cutoff if and only if its projection does. Based on joint work with Jonathan Hermon.

November 14, 2024: Nabarun Deb (University of Chicago)

Mean-Field fluctuations in Ising models and posterior prediction intervals in low signal-to-noise ratio regimes

Ising models have become central in probability, statistics, and machine learning. They naturally appear in the posterior distribution of regression coefficients under the linear model $Y = X\beta + \epsilon$, where $\epsilon \sim N(0, \sigma^2 I_n)$. This talk explores fluctuations of specific linear statistics under the Ising model, with a focus on applications in Bayesian linear regression.

In the first part, we examine Ising models on "dense regular" graphs and characterize the limiting distribution of average magnetization across various temperature and magnetization regimes, extending previous results beyond the Curie-Weiss (complete graph) case. In the second part, we analyze posterior prediction intervals for linear statistics in low signal-to-noise ratio (SNR) scenarios, also known as the contiguity regime. Here, unlike standard Bernstein-von Mises results, the limiting distributions are highly sensitive to the choice of prior. We illustrate this dependency by presenting limiting laws under both correctly specified and misspecified priors.

This talk is based on joint work with Sumit Mukherjee and Seunghyun Li.

November 21, 2024: Reza Gheissari (Northwestern)

Wetting and pre-wetting in (2+1)D solid-on-solid interfaces

The (d+1)D-solid-on-solid model is a simple model of integer-valued height functions that approximates the low-temperature interface of an Ising model. When $d\ge 2$, with zero-boundary conditions, at low temperatures the surface is localized about height $0$, but when constrained to take only non-negative values entropic repulsion pushes it to take typical heights of $O(\log n)$.  I will describe the mechanism of entropic repulsion, and present results on how the picture changes when one introduces a competing force trying to keep the interface localized (either an external field or a reward for points where the height is exactly zero). Along the way, I will outline rich predictions for the shapes of level curves, and for metastability phenomena in the Glauber dynamics. Based on joint work with Eyal Lubetzky and Joseph Chen.

November 28, 2024: Thanksgiving

No seminar

December 5, 2024: Erik Bates (NC State)

Parisi formulas in multi-species and vector spin glass models

The expression "Parisi formula" refers to a variational formula postulated by Parisi in 1980 to give the limiting free energy of the Sherrington--Kirkpatrick (SK) spin glass.  The SK model was originally conceived as a mean-field description for disordered magnetism, and has since become a mathematical prototype for frustrated disordered systems and high-complexity functions.  In recent years, there has been an effort to extend the Parisi framework to various generalizations of the SK model, raising new physical questions met with fresh mathematical challenges.  In this talk, I will share some developments in this evolving story.  Based on joint works with Leila Sloman and Youngtak Sohn.