## Seminars and Colloquia by Series

### TBA by Subhroshekhar Ghosh

Series
Stochastics Seminar
Time
Thursday, April 9, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Subhroshekhar GhoshNational University of Singapore

### Eigenvectors' overlaps for integrable models of non-Hermitian random matrices

Series
Stochastics Seminar
Time
Thursday, April 2, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Guillaume Dubach

Right and left eigenvectors of non-Hermitian matrices form a bi-orthogonal system to which one can associate homogeneous quantities known as overlaps. The matrix of overlaps quantifies the stability of the spectrum and characterizes the joint eigenvalues increments under Dyson-type dynamics. Overlaps first appeared in the physics literature: Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results can be obtained for quaternionic Gaussian matrices, as well as matrices from the spherical and truncated-unitary ensembles.

### Complexity of the pure spherical p-spin model

Series
Stochastics Seminar
Time
Thursday, March 12, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Julian GoldNorthwestern University

The pure spherical p-spin model is a Gaussian random polynomial H of degree p on an N-dimensional sphere, with N large. The sphere is viewed as the state space of a physical system with many degrees of freedom, and the random function H is interpreted as a smooth assignment of energy to each state, i.e. as an energy landscape.

In 2012, Auffinger, Ben Arous and Cerny used the Kac-Rice formula to count the average number of critical points of H having a given index, and with energy below a given value. This number is exponentially large in N for p > 2, and the rate of growth itself is a function of the index chosen and of the energy cutoff. This function, called the complexity, reveals interesting topological information about the landscape H: it was shown that below an energy threshold marking the bottom of the landscape, all critical points are local minima or saddles with an index not diverging with N. It was shown that these finite-index saddles have an interesting nested structure, despite their number being exponentially dominated by minima up to the energy threshold. The total complexity (considering critical points of any index) was shown to be positive at energies close to the lowest. Thus, at least from the perspective of the average number of critical points, these random landscapes are very non-convex. The high-dimensional and rugged aspects of these landscapes make them relevant to the folding of large molecules and the performance of neural nets.

Subag made a remarkable contribution in 2017, when he used a second-moment approach to show that the total number of critical points concentrates around its mean. In light of the above, when considering critical points near the bottom of the landscape, we can view Subag's result as a statement about the concentration of the number of local minima. His result demonstrated that the typical behavior of the minima reflects their average behavior. We complete the picture for the bottom of the landscape by showing that the number of critical points of any finite index concentrates around its mean. This information is important to studying associated dynamics, for instance navigation between local minima. Joint work with Antonio Auffinger and Yi Gu at Northwestern.

### Martingales and descents

Series
Stochastics Seminar
Time
Thursday, March 5, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Alperen OzdemirUniversity of Southern California

We provide a martingale proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order n^{-1/2}. The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics include inversions, descents in signed permutations, descents in Stirling permutations, the length of the longest alternating subsequences, descents in matchings and two-sided Eulerian numbers.

### Lower Deviations and Convexity

Series
Stochastics Seminar
Time
Thursday, February 27, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Petros ValettasUniversity of Missouri, Columbia

While deviation estimates above the mean is a very well studied subject in high-dimensional probability, for their lower analogues far less are known. However, it has been observed, in several key situations, that lower deviation inequalities exhibit very different and stronger behavior. In this talk I will discuss how convexity can serve as a key feature to (a) explain this distinction, (b) obtain improved lower tail bounds, and (c) characterize the tightness of Gaussian concentration.

### Critical first-passage percolation in high dimensions

Series
Stochastics Seminar
Time
Thursday, February 20, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jack HansonCity College of New York

In critical Bernoulli percolation on $\mathbb{Z}^d$ for $d$ large, it is known that there are a.s. no infinite open clusters. In particular, for n large, every path from the origin to the boundary of $[-n, n]^d$ must contain some closed edges. Let $T_n$ be the (random) minimal number of closed edges in such a path. How does $T_n$ grow with $n$? We present results showing that for d larger than the upper critical dimension for Bernoulli percolation ($d > 6$), $T_n$ is typically of the order $\log \log n$. This is in contrast with the $d = 2$ case, where $T_n$ grows logarithmically. Perhaps surprisingly, the model exhibits another major change in behavior depending on whether $d > 8$.

### Heat semigroup approach to isoperimetric inequalities in metric measure spaces

Series
Stochastics Seminar
Time
Thursday, January 30, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Patricia Alonso-RuizTexas A&amp;M University

The classical isoperimetric problem consists in finding among all sets with the same volume (measure) the one that minimizes the surface area (perimeter measure). In the Euclidean case, balls are known to solve this problem. To formulate the isoperimetric problem, or an isoperimetric inequality, in more general settings, requires in particular a good notion of perimeter measure.

The starting point of this talk will be a characterization of sets of finite perimeter original to Ledoux that involves the heat semigroup associated to a given stochastic process in the space. This approach put in connection isoperimetric problems and functions of bounded variation (BV) via heat semigroups, and we will extend these ideas to develop a natural definition of BV functions and sets of finite perimeter on metric measure spaces. In particular, we will obtain corresponding isoperimetric inequalies in this setting.

The main assumption on the underlying space will be a non-negative curvature type condition that we call weak Bakry-Émery and is satisfied in many examples of interest, also in fractals such as (infinite) Sierpinski gaskets and carpets. The results are part of joint work with F. Baudoin, L. Chen, L. Rogers, N. Shanmugalingam and A. Teplyaev.

### Probabilistic approach to Bourgain's hyperplane conjecture

Series
Stochastics Seminar
Time
Thursday, January 16, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Arnaud MarsigliettiUniversity of Florida

The hyperplane conjecture, raised by Bourgain in 1986, is a major unsolved problem in high-dimensional geometry. It states that every convex set of volume 1 in the Euclidean space has a section that is lower bounded away from 0 uniformly over the dimension. We will present a probabilistic approach to the conjecture.

### Learning mixtures of permutations from groups of comparisons

Series
Stochastics Seminar
Time
Thursday, January 9, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cheng MaoGeorgia Institute of Technology

In various applications involving ranking data, statistical models for mixtures of permutations are frequently employed when the population exhibits heterogeneity. In this talk, I will discuss the widely used Mallows mixture model. I will introduce a generic polynomial-time algorithm that learns a mixture of permutations from groups of pairwise comparisons. This generic algorithm, equipped with a specialized subroutine, demixes the Mallows mixture with a sample complexity that improves upon the previous state of the art.

### Zero-free regions and central limit theorems

Series
Stochastics Seminar
Time
Thursday, November 14, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Marcus MichelenUniversity of Illinois, Chicago

Let X be a random variable taking values in {0,...,n} and f(z) be its probability generating function.  Pemantle conjectured that if the variance of X is large and f has no roots close to 1 in the complex plane, then X must be approximately normal. We will discuss a complete resolution of this conjecture in a strong quantitative form, thereby giving the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. Additionally, if f has no roots with small argument, then X must be approximately normal, again in a sharp quantitative form. These results also imply a multivariate central limit theorem that answers a conjecture and completes a program of Ghosh, Liggett and Pemantle.  This talk is based on joint work with Julian Sahasrabudhe.