Workshop

The motivating question for this talk is: What does a sparse Erd\"os-R\'enyi random graph, conditioned to have twice the number of triangles than the expected number, typically look like? Motivated by this question, In 2014, Chatterjee and Dembo introduced a framework for obtaining Large Deviation Principles (LDP) for nonlinear functions of Bernoulli random variables (this followed an earlier work of Chatterjee-Varadhan which used limit graph theory to answer this question in the dense regime). The aforementioned framework relies on a notion of "low complexity" functions on the discrete cube, defined in terms of the covering numbers of their gradient. The central lemma used in their proof provides a method of estimating the log-normalizing constant $\log \sum_{x \in \{-1,1\}^n} e^{f(x)}$ by a corresponding mean-field functional. In this talk, we will introduce a new notion of complexity for measures on the discrete cube, namely the mean-width of the gradient of the log-density. We prove a general structure theorem for such measures which goes beyond the discrete cube. In particular, we show that a measure $\nu$ attaining low complexity (with no extra smoothness assumptions needed) are close to a product measure in the following sense: there exists a measure $\tilde \nu$ a small "tilt" of $\nu$ in the sense that their log-densities differ by a linear function with small slope, such that $\tilde \nu$ is close to a product measure in transportation distance. An easy corollary of our result is a strengthening of the framework of Chatterjee-Dembo, which in particular simplifies the derivation of LDPs for subgraph counts, and improves the attained bounds. We will demonstrate how our framework can be used to study the behavior of low-complexity measures beyond the approximation of the partition function. As an example application, we prove that exponential random graphs behave roughly like mixtures of stochastic block models.

Optimal matching problems are random variational problems widely investigated in the mathematics and physics literature. We discuss here the optimal matching problem of an empirical measure on a sample of iid random variables to the common law in Kantorovich-Wasserstein distances, which is a classical topic in probability and statistics. Two-dimensional matching of uniform samples gave rise to deep results investigated from various view points (combinatorial, generic chaining). We study here the case of Gaussian samples, first in dimension one on the basis of explicit representations of Kantorovich metrics and a sharp analysis of more general log-concave distributions in terms of their isoperimetric profile (joint work with S. Bobkov), and then in dimension two (and higher) following the PDE and transportation approach recently put forward by L. Ambrosia, F. Stra and D. Trevisan.

We will be discussing strong distances between probability distributions given by the Renyi divergences. 

The main focus is put on the study of the asymptotic behavior along normalized convolutions and the associated central limit theorem.

We will discuss the fluctuations in discrete Beta-ensembles. Such distributions describe for instance the positions of lozenge tilings and ressemble the continuous Beta-ensembles which appear in random matrix ensembles. We will highlight the role of equations introduced by Nekrasov in this context, which play the same crucial role than Dyson-Schwinger equations in the continuous case, but are not obtained by a straightforward integration by part.

The mathematical analysis of Coulomb gases, especially in dimensions higher than one, has been the focus of much recent activity. For the 3D Coulomb, there is a famous prediction of Jancovici, Lebowitz and Manificat that if N is the number of particles falling in a given region, then N has fluctuations of order cube-root of E(N). I will talk about the recent proof of this conjecture for a closely related model, known as the 3D hierarchical Coulomb gas. I will also try to explain, through some toy examples, why such unusually small fluctuations may be expected to appear in interacting gases

In this talk we will discuss the following analog of Bezout inequality for mixed volumes: $$ V(P_1,\dots,P_r,\Delta^{n-r})V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(P_i,\Delta^{n-1})\ \text{ for }2\leq r\leq n. $$ We will show that the above inequality is true when $\Delta$ is an $n$-dimensional simplex and $P_1, \dots, P_r$ are convex bodies in ${\mathbb R}^n$. We present a conjecture that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $\Delta$ must be an $n$-dimensional simplex. We will show that the conjecture is true for many special cases, for example, in ${\mathbb R}^2$ or if we assume that $\Delta$ is a convex polytope. Next we will discuss an isomorphic version of the Bezout inequality: what is the best constant $c(n,r)>0$ such that $$ V(P_1,\dots,P_r,\Delta^{n-r})V_n(D)^{r-1}\leq c(n,r)\prod_{i=1}^r V(P_i,D^{n-1})\ \text{ for }2\leq r\leq n, $$ where $P_1, \dots, P_r, D$ are convex bodies in ${\mathbb R}^n.$ Finally, we will present a connection of the above inequality to inequalities on the volume of orthogonal projections of convex bodies as well as inequalities for zonoids.

Different approaches to introduce intrinsic volumes and more generally mixed volumes for convex and log-concave functions were proposed by Bobkov, Colesanti and Fragal\` a, by Rotem and Milman and by Alesker. They all turn out to be valuations on the corresponding spaces. Here a new class of continuous valuations on the space of convex functions on ${\mathbb R}^n$ is introduced. On smooth convex functions, they are defined for $i=0,\dots,n$ by \begin{equation*} u\mapsto \int_{{\mathbb R}^n} \zeta(u(x),x,\nabla u(x))\,[{{D}^2} u(x)]_i\,d x \end{equation*} where $\zeta\in C({\mathbb R}\times{\mathbb R}^n\times{\mathbb R}^n)$ and $[{{D}^2} u]_i$ is the $i$-th elementary symmetric function of the eigenvalues of the Hessian matrix, ${{D}^2} u$, of $u$. Under suitable assumptions on $\zeta$, these valuations are shown to be invariant under translations and rotations on convex and coercive functions. Ultimately, a complete classification of continuous and rigid motion invariant valuations on this space of functions is the aim of this approach. The connection to Hadwiger's theorem will be discussed. The results presented in this talk are joint with Andrea Colesanti (University of Florence) and Fabian Mussnig (Technische Universit\"at Wien).

We derive a lower bound on the smallest singular value of a random $d$-regular matrix, that is, the adjacency matrix of a random $d$-regular directed graph. More precisely, let $C_1<d< c_1 n/\log^2 n$ and let $M$ be uniformly distributed on the set of all $0/1$-valued $n\times n$ matrices such that each row and each column of a matrix has exactly $d$ ones. Then the smallest singular value $s_{n} (M)$ of $M$ is greater than $c_2 n^{-6}$ with probability at least $1-C_2\log^2 d/\sqrt{d}$, where $c_1$,  $c_2$, $C_1$, and $C_2$ are absolute positive constants. This is a joint work with A. Lytova, K. Tikhomirov, N. Tomczak-Jaegermann, and P. Youssef

I will discuss the connections between weak transportation inequalities with quadratic-linear cost functions, Bobkov-Ledoux type modified log-Sobolev inequalities for convex/concave functions and the convex Poincare inequality. If time permits I will also mention some self normalized moment inequalities for general convex functions, which can be easily obtained from concentration estimates for Lipschitz convex functions. Based on joint work with Michal Strzelecki.

Let A be an n by n random matrix with independent centered rows, so that each row has real-valued components, is isotropic and log-concave. Further, let M be any fixed n by n real matrix. We derive small ball probability estimates for the smallest singular value of the non-centered random matrix A+M. Our method is free from any use of covering arguments, and is principally different from a standard (by now) approach involving a decomposition of the unit sphere and coverings, as well as from an approach of Sankar-Spielman-Teng for non-centered Gaussian matrices. Our method gives an estimate for the condition number of A+M which essentially matches the known bounds for a non-centered Gaussian matrix.

The classical Loomis-Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide upper bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer's dual Loomis-Whitney inequality. Joint work with S. Brazitikos and D-M. Liakopoulos. We also discuss applications of these inequalities to questions regarding the surface area of lower dimensional projections and the average section functional of lower dimensional sections of a convex body, which were the motivation for them (joint works with A. Koldobsky, S. Dann, S. Brazitikos and P. Valettas).