Current Series

[View Past Series]

Fri Sep 14

Probability Seminar

2:30pm - Vincent Hall 209
The travel time to infinity in percolation
Wai-Kit Lam, UMN

On the two-dimensional square lattice, assign i.i.d. nonnegative weights to the edges with common distribution $F$. For which distributions $F$ is there an infinite self-avoiding path with finite total weight? It has long been known that there is no such infinite path when $F(0) < 1/2$ (there are only finite clusters of zero-weight edges), and there is one when $F(0) > 1/2$ (there is an infinite cluster of zero-weight edges). The critical case, $F(0) = 1/2$, is considerably more difficult due to the presence of finite clusters of zero-weight edges on all scales. In a joint work with M. Damron and X. Wang, we give a necessary and sufficient condition on $F$ for the existence of an infinite finite-weight path, and study the asymptotic behaviors of the first-passage time in the critical case. I will also discuss a recent work with M. Damron and J. Hanson in which we prove some limit theorems for the first-passage time in the critical case on the triangular lattice.

Fri Sep 21

Probability Seminar

2:30pm - Vincent Hall 209
The maximum of the characteristic polynomial for a random permutation matrix
Nicholas Cook, UCLA

Let $P$ be a uniform random permutation matrix of size $N$ and let $\chi_N(z)= \det(zI - P)$ denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of $\chi_N$ on the unit circle, specifically,
\sup_{|z|=1}|\chi_N(z)|= N^{x_c + o(1)}
with probability tending to one as $N\to \infty$, for a numerical constant $x_c\approx 0.652$. The main idea of the proof is to uncover an approximate branching structure in the distribution of (the logarithm of) $\chi_N$, viewed as a random field on the circle, and to adapt a well-known second moment argument for the maximum of the branching random walk. Unlike the well-studied \emph{CUE field} in which $P_N$ is replaced with a Haar unitary, the distribution of $\chi_N(z)$ is sensitive to Diophantine properties of the argument of $z$. To deal with this we borrow tools from the Hardy--Littlewood circle method in analytic number theory. Based on joint work with Ofer Zeitouni.

Fri Sep 28

Probability Seminar

2:30pm - Vincent Hall 209
Stationary coalescing walks on the lattice
Arjun Krishnan, University of Rochester

Consider a measurable dense family of semi-infinite nearest-neighbor paths on the integer lattice in d dimensions. If the measure on the paths is translation invariant, we completely classify their collective behavior in d=2 under mild assumptions. We use our theory to classify the behavior of semi-infinite geodesics in random translation invariant metrics on the lattice; it applies, in particular, to first- and last-passage percolation. We also construct several examples displaying unexpected behaviors. (joint work with Jon Chaika)

Fri Oct 12

Probability Seminar

2:30pm - Vincent Hall 209
Brascamp-Lieb inequalities for even functions
Liran Rotem, UMN

It was observed by Cordero-Erausquin, Fradelizi and Maurey that the classical Gaussian Poincare Inequality can be improved by a factor of 2 if we restrict our attention to even functions. They used this improved inequality to answer a geometric question asked by Banaszczyk about convexity of the Gaussian measure.

The Gaussian Poincare inequality is a special case of a more general variance inequality due to Brascamp and Lieb. In general, it is not clear how to improve this more general inequality in the even case. Again, this question turns out to be closely related to certain geometric problems. In this talk we will prove a sharp even Brascamp-Lieb inequality for measures of the form exp(-|x|^p). To do so we will also present a new weighted Poincare inequality which holds for ODD functions.

Based on joint work with Dario Cordero-Erausquin.

Fri Oct 26

Probability Seminar

2:30pm - Vincent Hall 209
Some Problems of Robust High-dimensional Statistics
Gilad Lerman, University of Minnesota

The talk will first review the problem of robust subspace recovery, which seeks an underlying low-dimensional subspace in a data set that is possibly corrupted with outliers. The emphasis will be on surveying existing theoretical guarantees and tradeoffs. New results for adversarial outliers will also be mentioned. Following this, other related problems will be discussed, along with new results for one of these problems.

Fri Nov 02

Probability Seminar

2:30pm - Vincent Hall 209
Dynamical freezing in a spin glass system with logarithmic correlations
Julian Gold, Northwestern University

We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. Joint work with Aser Cortines (University of Zurich) and Oren Louidor (Technion).

Fri Nov 09

Probability Seminar

2:30pm - Vincent Hall 209
Low-temperature localization of directed polymers
Erik Bates, Stanford University

On the d-dimensional integer lattice, directed polymers are paths of a random walk that have been reweighted according to a random environment that refreshes at each time step. The qualitative behavior of the system is governed by a temperature parameter; if this parameter is small, the environment has little effect, meaning all possible paths are close to equally likely. If the parameter is made large, however, the system undergoes a phase transition at which the path’s endpoint starts to localize. To understand the extent of this localization, we exploit the underlying Markov structure of the quenched endpoint distribution. The key difficulty is that the space of measures is too large for one to expect convergence results. By adapting methods appearing in the work of Mukherjee and Varadhan, we develop a compactification theory to resolve the issue. In this talk, we will discuss this intriguing abstraction, as well as new concrete theorems it allows us to prove for directed polymers constructed from SRW or any other walk. (joint work with Sourav Chatterjee)

Fri Nov 16

Probability Seminar

2:30pm - Vincent Hall 209
Macroscopic fluctuations through Schur generating functions
Vadim Gorin, MIT

I will talk about a special class of large-dimensional stochastic systems with strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.
It is believed that macroscopic fluctuations in such systems are universally described by log-correlated Gaussian fields. I will present an approach to handle this question based on the notion of the Schur generating function of a probability distribution, and explain how it leads to a rigorous confirmation of this belief in a variety of situations.

Fri Dec 07

Probability Seminar

2:30pm - Vincent Hall 209
When particle systems meet PDEs
Li-Cheng Tsai, Columbia University

Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three paradigmatic facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.

Fri Jan 25

Probability Seminar

2:30pm - Vincent Hall 213
Large deviations for sparse random graphs
Nicholas Cook, Stanford University

Let $G=G(N,p)$ be an Erd\H{o}s--R\'enyi graph on $N$ vertices (where each pair is connected by an edge independently with probability $p$). We view $N$ as going to infinity, with $p$ possibly going to zero with $N$. What is the probability that $G$ contains twice as many triangles as we would expect? I will discuss recent progress on this ``infamous upper tail" problem, and more generally on tail estimates for counts of any fixed subgraph. These problems serve as a test bed for the emerging theory of \emph{nonlinear large deviations}, and also connect with issues in extending the theory of \emph{graph limits} to handle sparse graphs. In particular, I will discuss our approach to the upper tail problems via new versions of the classic regularity and counting lemmas from extremal combinatorics, specially tailored to the study of random graphs in the large deviations regime. This talk is based on joint work with Amir Dembo.

Fri Feb 01

Probability Seminar

2:30pm - Vincent Hall 213
Nonequilibrium particle systems in inhomogeneous space
Leonid Petrov, University of Virginia

I will discuss stochastic interacting particle systems in the Kardar-Parisi-Zhang universality class evolving in one-dimensional inhomogeneous space. The inhomogeneity means that the speed of a particle depends on its location. I will focus on integrable cases, i.e., for which certain observables can be written in an exact form suitable for asymptotic analysis. Examples include a new continuous-space version of TASEP (totally asymmetric simple exclusion process), and the pushTASEP (=long-range TASEP). For integrable systems, limit shapes of particles density can be described in an explicit way. Asymptotics of fluctuations around infinite traffic jams, also available explicitly, bring phase transitions of a novel nature.

Fri Feb 08

Probability Seminar

2:30pm - Vincent Hall 213
Stability of the Bakry-Emery theorem on R^n
Tom Courtade, UC Berkeley

In R^n, the Bakry-Émery theorem states that the sharp constant in various functional inequalities for uniformly log-concave measures is no worse than the sharp constant for the Gaussian measure. As a consequence, uniformly log-concave measures inherit certain nice features of the Gaussian, like good measure concentration properties.

In this talk, I'll discuss quantitative stability estimates for the Bakry-Émery bound on logarithmic Sobolev and Poincaré constants. In particular, if a 1-uniformly log-concave measure has almost the same logarithmic Sobolev or Poincaré constant as the standard Gaussian measure, then it must (almost) split off a Gaussian factor. As a corollary, we obtain dimension-free stability estimates for Gaussian concentration of Lipschitz functions. The proofs combine Stein’s method, optimal transport, and simple variational arguments.

Joint work with Max Fathi.

Fri Feb 22

Probability Seminar

2:30pm - Vincent Hall 213
Largest Entries of Sample Correlation Matrices from Equi-correlated Normal Populations
Tiefeng Jiang, UMN

We the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the entries of the population distribution have a common correlation coefficient r >0 and both the population dimension p and the sample size n tend to infinity with log p=o(n^{1/3}).
As 0< r<1, we prove that the largest off-diagonal entry of the sample correlation matrix converges to a Gaussian distribution, and the same is true for the sample covariance matrix as 0< r<1/2. This differs substantially from a well-known result for the independent case where r=0, in which the above limiting distribution is an extreme-value distribution. We then study the phase transition between these two limiting distributions and identify the regime of r where the transition occurs. It turns out that the thresholds of such a regime depend on n and converge to zero. If r is less than the threshold, larger than the threshold or is equal to the threshold, the corresponding limiting distribution is the extreme-value distribution, the Gaussian distribution and a convolution of the two distributions, respectively. The proofs rely on a subtle use of the Chen-Stein Poisson approximation method, conditioning, a coupling to create independence and a special property of sample correlation matrices. The results are then applied to evaluating the power of a high-dimensional testing problem of identity correlation matrix.

Fri Mar 15

Probability Seminar

2:30pm - Vincent Hall 213
The generalized TAP free energy
Wei-Kuo Chen, UMN

Spin glasses are disordered spin systems initially invented by theoretical physicists with the aim of understanding some strange magnetic properties of certain alloys. In particular, over the past decades, the study of the Sherrington-Kirkpatrick (SK) mean-field model via the replica method has received great attention. In this talk, I will discuss another approach to studying the SK model proposed by Thouless-Anderson-Palmer (TAP). I will explain how the generalized TAP correction appears naturally and give the corresponding generalized TAP representation for the free energy. Based on a joint work with D. Panchenko and E. Subag.

Fri Mar 29

Probability Seminar

2:30pm - Vincent Hall 213
Moments of scores
Sergey Bobkov, UMN

If a random variable X has an absolutely continuous
density f, its score is defined to be the random variable
\rho(X) = f'(X)/f(X), where f' is the derivative of f. We will discuss
upper bounds on the moments of the scores, especially in the
case when X represents the sum of independent random variables.

Fri Apr 05

Probability Seminar

2:30pm - Vincent Hall 213
Frog model on trees with drift
Si Tang, Lehigh University

We provide a uniform upper bound on the minimal drift so that the one-per-site frog model on a d-ary tree is recurrent. To do this, we introduce a subprocess that couples across trees with different degrees. Finding couplings for frog models on nested sequences of graphs is known to be difficult. The upper bound comes from combining the coupling with a new, simpler proof that the frog model on a binary tree is recurrent when the drift is sufficiently strong. This is a joint work with E. Beckman, N. Frank, Y. Jiang, and M. Junge.

Fri Apr 19

Probability Seminar

2:30pm - Vincent Hall 213
Optimal Spectral Shrinkage and PCA with Heteroscedastic Noise
Will Leeb, UMN

I will present recent results on the related problems of denoising, covariance estimation, and principal component analysis for the spiked covariance model with heteroscedastic noise. Specifically, I will present an estimator of the principal components based on whitening the noise, and optimal spectral shrinkers for use with these estimated principal components. I will also show new results on the optimality of whitening for principal subspace estimation. This is joint work with Elad Romanov of the Hebrew University.

Fri Apr 26

Probability Seminar

2:30pm - Vincent Hall 213
Majority vote processes on trees
Maury Bramson, UMN

The majority vote process was one of the first interacting particle systems to be investigated and can be described as follows. There are two possible opinions at each site and that opinion switches randomly to the majority opinion of the neighboring sites. Also, at a different rate epsilon, the opinion at each site randomly changes due to noise.

Despite its simple dynamics, the majority vote process is difficult to analyze. In particular, on Z^d with d>1 and epsilon chosen small, it is not known whether there exists more than one equilibrium. This is surprising due to the close analogy between the majority vote process and the Ising model.

Here, we discuss work with Larry Gray on the majority vote process on the infinite tree with vertex degree d, where it is shown that, for small noise, there are uncountably many mutually singular equilibria, and that convergence to equilibrium occurs exponentially quickly from nearby initial states. Our methods rely on graphical constructions; they are quite flexible and can be used to obtain analogous results for other models, such as the stochastic Ising model on a tree.

This is the first part of a two-lecture presentation, and will concentrate on background and on existence of equilibria. Larry Gray will give the second part the following week, which will concentrate on convergence.

Fri May 03

Probability Seminar

2:30pm - Vincent Hall 213
The majority vote process on trees: convergence to equilibrium
Larry Gray, UMN

This is joint with Maury Bramson, who introduced our work last week and then spoke more particularly about the existence of uncountably many mutually singular equilibria in the majority vote process on a tree of vertex degree 5 and higher, whenever the noise rate is sufficiently small. Each equilibrium is associated with a different fixed point of the majority vote operation.

Natural question: If the system starts at (or near) one of these fixed points and the noise rate is small, does it converge to the corresponding equilibrium, and if so, how fast? Answering this question can tell us much about the properties of these equilibria, but standard techniques do not apply. I will describe a new method, based on the "graphical construction" of these systems, which shows exponentially quick convergence to equilibrium when the noise rate is sufficiently small. These methods apply to other models on trees, such as the stochastic Ising model, for which the convergence result was previously unknown.

This lecture is self-contained; essential features from the previous week will be included (with some new pictures).