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[[Probability | Back to Probability Group]]


= Fall 2020 =
* '''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>
 
<b> IMPORTANT: </b> In Fall 2020 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].
== September 17, 2020, [https://www.math.tamu.edu/~bhanin/ Boris Hanin] (Princeton and Texas A&M) ==
 
'''Pre-Talk: (1:00pm)'''
 
'''Neural Networks for Probabilists''' 
 
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.
 
'''Talk: (2:30pm)'''
 
'''Effective Theory of Deep Neural Networks'''
 
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.
 
== September 24, 2020, [https://people.ucd.ie/neil.oconnell Neil O'Connell] (Dublin)  ==


'''Some new perspectives on moments of random matrices'''


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


== October 1, 2020, [https://marcusmichelen.org/ Marcus Michelen] (UIC) ==
We usually end for questions at 3:20 PM.


'''Roots of random polynomials near the unit circle'''
== September 5, 2024: ==
No seminar


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.
== September 12, 2024: Hongchang Ji (UW-Madison) ==
'''Spectral edge of non-Hermitian random matrices'''


== October 8, 2020, [http://sites.harvard.edu/~sus977/index.html Subhabrata Sen] (Harvard) ==
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.


'''Large deviations for dense random graphs: beyond mean-field'''
Based on joint works with A. Campbell, G. Cipolloni, and L. Erd\H{o}s.


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.


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
== September 19, 2024: Miklos Racz (Northwestern) ==
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.
'''The largest common subtree of uniform attachment trees'''


Based on joint works with Christian Borgs, Jennifer Chayes, Souvik Dhara, Julia Gaudio and Samantha Petti.
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.


== October 15, 2020, [https://math.cornell.edu/philippe-sosoe Philippe Sosoe] (Cornell) ==
== September 26, 2024: Dmitry Krachun (Princeton) ==
'''A glimpse of universality in critical planar lattice models'''


Title: '''Concentration in integrable polymer 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.


I will discuss a general method, applicable to all known integrable stationary polymer models, to obtain nearly optimal bounds on the
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.  
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.
Based on joint works with Hugo Duminil-Copin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, and Tatiana Tikhonovskaia.


==October 22, 2020, [http://www.math.toronto.edu/balint/ Balint Virag] (Toronto) ==
== October 3, 2024: Joshua Cape (UW-Madison) ==
'''A new random matrix: motivation, properties, and applications'''


Title: '''The heat and the landscape'''
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.


Abstract: The directed landscape is the conjectured universal scaling limit of the
== October 10, 2024: Midwest Probability Colloquium ==
most common random planar metrics. Examples are planar first passage
N/A
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
== October 17, 2024: Kihoon Seong (Cornell) ==
Peche statistics well-known from random matrix theory.
'''Gaussian fluctuations of focusing Φ^4 measure around the soliton manifold'''


This provides a general and elementary method for showing convergence to
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).  
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.
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.


==October 29, 2020, [https://www.math.wisc.edu/node/80 Yun Li] (UW-Madison) ==
This talk is based on joint work with Philippe Sosoe (Cornell).


Title: '''Operator level hard-to-soft transition for β-ensembles'''
== October 24, 2024: Jacob Richey (Alfred Renyi Institute) ==
'''Stochastic abelian particle systems and self-organized criticality'''


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.
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.


== November 5, 2020, [http://sayan.web.unc.edu/ Sayan Banerjee] (UNC at Chapel Hill) ==
== October 31, 2024: David Clancy (UW-Madison) ==
'''Likelihood landscape on a known phylogeny'''


Title: '''Persistence and root detection algorithms in growing networks'''
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.


Abstract: Motivated by questions in Network Archaeology, we investigate statistics of dynamic networks
This talk is based on joint work with Hanbaek Lyu, Sebastien Roch and Allan Sly.
that are ''persistent'', that is, they fixate almost surely after some random time as the network grows. We
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.
== November 7, 2024: Zoe Huang (UNC Chapel Hill) ==
'''Cutoff for Cayley graphs of nilpotent groups'''


== November 12, 2020, [https://cims.nyu.edu/~ajd594/ Alexander Dunlap] (NYU Courant Institute) ==
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.


Title: '''A forward-backward SDE from the 2D nonlinear stochastic heat equation'''
== 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


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).
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.


== November 19, 2020, [https://statistics.wharton.upenn.edu/profile/dingjian/ Jian Ding] (University of Pennsylvania) ==
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.


Title: '''Correlation length of two-dimensional random field Ising model via greedy lattice animal'''
This talk is based on joint work with Sumit Mukherjee and Seunghyun Li.


Abstract: In this talk, I will discuss two-dimensional random field Ising model where the disorder is given by i.i.d. mean zero Gaussian variables with small variance. In particular, I will present a recent joint work with Mateo Wirth on (one notion of) the correlation length, which is the critical size of the box at which the influences to spin magnetization from the boundary conditions and from the random field are comparable. Our work draws a connection to the greedy lattice animal normalized by the boundary size.
== November 21, 2024: Reza Gheissari (Northwestern) ==
'''Wetting and pre-wetting in (2+1)D solid-on-solid interfaces'''


== December 3, 2020, [https://www.math.wisc.edu/people/faculty-directory Tatyana Shcherbina] (UW-Madison) ==
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.


Title: '''SUSY transfer matrix approach for the real symmetric 1d random band matrices '''
== November 28, 2024: Thanksgiving ==
No seminar


Abstract: Random band matrices (RBM) are natural intermediate models to study
== December 5, 2024: Erik Bates (NC State) ==
eigenvalue statistics and quantum propagation in disordered systems,
since they interpolate between mean-field type Wigner matrices and
random Schrodinger operators. In particular, RBM can be used to model the
Anderson metal-insulator phase transition. The conjecture states that the eigenvectors
of $N\times N$ RBM are completely delocalized and the local spectral statistics governed 
by  the Wigner-Dyson statistics for large bandwidth $W$ (i.e. the local behavior is
the same as for Wigner matrices), and by Poisson statistics for a small $W$
(with exponentially localized eigenvectors). The transition is conjectured to
be sharp and for RBM in  one  spatial  dimension  occurs around  the  critical 
value $W=\sqrt{N}$. Recently, we proved the universality of the correlation
functions for the whole delocalized region $W\gg \sqrt{N}$ for a certain type
of Hermitian Gaussian RBM. This result was obtained by
application of the supersymmetric method (SUSY) combined with the transfer matrix approach.
In this talk I am going to discuss how this techniques can be adapted to the
real symmetric case.


== December 10, 2020, [https://www.ewbates.com/ Erik Bates] (UW-Madison) ==
'''Parisi formulas in multi-species and vector spin glass models'''


Title: '''Empirical measures, geodesic lengths, and a variational formula in first-passage percolation'''
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.
 
Abstract: We consider the standard first-passage percolation model on $\mathbb{Z}^d$, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them. Our primary interest is in the empirical measures of edge-weights observed along geodesics from $0$ to $n\mathbf{e}_1$.  For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as $n$ tends to infinity.  The key tool is a new variational formula for the time constant. In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.
 
 
[[Past Seminars]]

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.

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