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[[Probability | Back to Probability Group]]
[[Probability | Back to Probability Group]]


= Fall 2022 =
* '''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 at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>
[[Past Seminars]]


We usually end for questions at 3:20 PM.


[https://uwmadison.zoom.us/j/91828707031?pwd=YUJXMUJkMDlPR0VRdkRCQVJtVndIdz09 ZOOM LINK. Valid only for online seminars.]
= Fall 2024 =
<b>Thursdays at 2:30 PM either in 901 Van Vleck Hall or on Zoom</b>


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].
We usually end for questions at 3:20 PM.
 
 
== 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'''
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)    ==
== September 5, 2024: ==
No seminar


'''Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees'''
== September 12, 2024: Hongchang Ji (UW-Madison) ==
'''Spectral edge of non-Hermitian random matrices'''


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.  
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.
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: [https://danielslonim.github.io/ Daniel Slonim] (University of Virginia)  ==
Based on joint works with A. Campbell, G. Cipolloni, and L. Erd\H{o}s.


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


== 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)  ==
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.


'''"In law" ergodic theorem for the environment viewed from Sinaï's walk'''
== September 26, 2024: Dmitry Krachun (Princeton) ==
'''A glimpse of universality in critical planar lattice models'''  


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
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.  
depending on the steps <math>\scriptsize\Delta X_k:=X_{k+1}-X_k</math>.


== October 20, 2022, '''4pm, VV911''', in person: [https://tavarelab.cancerdynamics.columbia.edu/ Simon Tavaré] (Columbia University) ==
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.  
''Note the unusual time and room!''


'''An introduction to counts-of-counts data'''
Based on joint works with Hugo Duminil-Copin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, and Tatiana Tikhonovskaia.


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
== October 3, 2024: Joshua Cape (UW-Madison) ==
'''A new random matrix: motivation, properties, and applications'''


''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.
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.


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.
== 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'''


''References''
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).


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


[2] Arratia R, Barbour AD & Tavaré S. ''Logarithmic Combinatorial Structures,'' EMS, 2002
This talk is based on joint work with Philippe Sosoe (Cornell).


[3] Ewens WJ. Theoret Popul Biol, 3, 1972
== October 24, 2024: Jacob Richey (Alfred Renyi Institute) ==
'''Stochastic abelian particle systems and self-organized criticality'''


[4] Da Silva P, Jamshidpey A, McCullagh P & Tavaré S. Bernoulli Journal, in press, 2022 (online)  
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 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) ==  
== October 31, 2024: David Clancy (UW-Madison) ==
'''Likelihood landscape on a known phylogeny'''


'''Maximum weight matching on sparse graphs'''
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.


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


This is joint work with Wai-Kit Lam.
== November 7, 2024: Zoe Huang (UNC Chapel Hill) ==
'''Cutoff for Cayley graphs of nilpotent groups'''


== November 3, 2022, in person: [https://www.ias.edu/scholars/sky-yang-cao Sky Cao] (Institute for Advanced Study)   ==
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.


'''Exponential decay of correlations in finite gauge group lattice gauge theories'''
== 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


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.
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 10, 2022, in person: [https://math.wisc.edu/staff/clancy-david/ David Clancy] (UW-Madison)  ==
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.


'''Component Sizes of the degree corrected stochastic blockmodel'''
This talk is based on joint work with Sumit Mukherjee and Seunghyun Li.


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 21, 2024: Reza Gheissari (Northwestern) ==
'''Wetting and pre-wetting in (2+1)D solid-on-solid interfaces'''


== 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)  ==
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.


'''Integration by Parts and the KPZ Two-Point Function'''
== November 28, 2024: Thanksgiving ==
No seminar


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 5, 2024: Erik Bates (NC State) ==


== December 1, in person: [https://cims.nyu.edu/~ajd594/ Alex Dunlap] (Courant Institute)  ==
'''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.
== December 8, 2022, in person: [https://sites.northwestern.edu/juliagaudio/ Julia Gaudio] (Northwestern University)  ==
 
 
[[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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