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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.