# Seminars & Events for Ergodic Theory & Statistical Mechanics

##### Dynamics of 2D point vortics and its generalizations

##### Shortest Curves Associated to a Degenerate Jacobi Metric on the two Torus

Let $P$ be a potential on the two torus that takes its minimum value at a unique point m. Set $E_0 := P(m)$. For a real number $E$, let $g_E$ be the Jacobi metric associated to $P$ and $E$. For $E \gt E_0, g_E$ is a Riemannian metric. An ancient theorem of Morse and Hedlund says that a $g_E$-shortest curve in an indivisible homology class is simple. For $E = E_0, g_E$ is no longer a Riemannian metric because it vanishes at $m$. (It is a Riemannian metric in the complement of $m$.) For a suitable potential $P$, and a suitable indivisible homology class $h$, a $g_{E_0}$-shortest curve in $h$ crosses itself at $m$, so the theorem of Hedlund and Morse does not generalize to the case $E = E_0$.

##### The Spectrum of an Hermitian Matrix With Dependent Entries Constructed from Random Independent Images

In this talk we will present a preliminary analysis and numerical results for the distribution of eigenvalues of a certain random N by N Hermitian matrix, whose construction is motivated by a problem in structural biology. The matrix is built from N images, where each image is an array of P pixels, and the pixels are i.i.d standard Gaussians. Numerical experiments suggest that the spectrum approaches Wigner's semi-circle law for P>>N, but differs significantly from the semi-circle for P<<N. We attribute this difference to the dependencies among the matrix entries. In particular, using the third moment, we show that for fixed P, the largest eigenvalue is O(N) as N goes to infinity.

##### From random tilings to representation theory

Lozenge tilings of planar domains provide a simple, yet sophisticated model of random surfaces. Asymptotic behavior of such models has been extensively studied in recent years.

We will start from recent results about q-distributions on tilings of a hexagon or, equivalently, on boxed plane partitions. (This part is based on the joint work with A.Borodin and E.Rains).

In the second part of the talk we will explain how representation theory of the infinite-dimensional unitary group is related to random lozenge tilings with a certain Gibbs property. We will discuss applications of this correspondence and results on the classification of Gibbs measures on tilings of the half-plane.

##### Diffusive or superdiffusive asymtotics for periodic and non-periodic Lorentz processes

After the first success in establishing the diffusive, Brownian limit of planar, finite-horizon, periodic Lorentz processes, in 1981 Sinai turned the interest toward studying models when periodicity is hurt, in particular, to locally perturbed Lorentz processes. The 1981 solution for a random-walk-model - by Telcs and the speaker - only led in 2009 to that for the locally perturbed, finite-horizon Lorentz process (by Dolgopyat, Varju and the present author). Beside reporting on these results we also analyze the first steps in extending the limits obtained for the periodic Lorentz process to locally perturbed periodic or quasi-periodic ones (results by Nandori, Paulin, Varju and the speaker).

##### A survey of results in universality of Wigner matrices, Part I

In the 1950's, Wigner proved the famous semicircle laws for Wigner matrices and started the study of universality results in random matrices. In these two talks, this will serve as our starting point as we surveyed the historical developments in this field. We will end with a discussion of the proof of the local semicircle law of Erdos, Schlein and Yau and the four moment theorem by Tao and Vu. We will also discuss some of the open problems in the study of random matrices if time permits.

##### A survey of results in universality of Wigner matrices, Part II

In the 1950's, Wigner proved the famous semicircle laws for Wigner matrices and started the study of universality results in random matrices. In these two talks, this will serve as our starting point as we surveyed the historical developments in this field. We will end with a discussion of the proof of the local semicircle law of Erdos, Schlein and Yau and the four moment theorem by Tao and Vu. We will also discuss some of the open problems in the study of random matrices if time permits.

##### Two results on rigidity of commutative actions by toral automorphisms

In 1983 Berend proved rigidity of higher-rank commutative actions by toral automorphisms under some hyperbolicity and irreduciblity assumptions. We will present two rigidity results that respectively extend Berend's theorem to certain non-hyperbolic and reducible cases. We will also discuss some counterexamples of non-homogeneous orbit closures. This is joint work with Elon Lindenstrauss.

##### On the distribution of gaps for saddle connection directions

In joint work with J. Chaika, we prove results on the distribution of gaps of angles between saddle connections on flat surfaces. Our techniques draw on the work of Marklof-Strombergsson on the periodic Lorentz gas and that of Eskin-Masur on flat surfaces. We describe some applications to billiards in polygons.

##### Discrete Schrödinger operators with periodic and almost periodic potentials

The first half of the talk will be a survey; I will start from general facts about discrete Schrödinger operators with periodic potentials, and then discuss operators with almost periodic potentials, focusing on the almost Mathieu operator. In the second half, I will state and prove some results connecting the absolutely continuous spectrum of almost periodic operators to the spectra of their periodic approximations.

##### Effective Limit Distribution of the Frobenius Numbers

The Frobenius number of a lattice point $\bf{a}$ with positive coprime coordinates, is the largest integer which can NOT be expressed as a non-negative integer linear combination of the coordinates of $\bf{a}$. Marklof showed in 2010 that the limit distribution of the Frobenius numbers is given by the distribution for the covering radius function of a random unimodular lattice. The aim of the talk is to discuss the reason of this phenomenon, and indicate how to obtain the rate of the convergence of the corresponding distribution functions.

##### Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions

We consider the solution of the stochastic heat equation with multiplicative noise and delta function initial condition whose logarithm, with appropriate normalizations, is the free energy of the continuum directed polymer, or the solution of the Kardar-Parisi-Zhang equation with narrow wedge initial conditions. We prove explicit formulas for the one-dimensional marginal distributions — the crossover distributions — which interpolate between a standard Gaussian distribution (small time) and the GUE Tracy-Widom distribution (large time). The proof is via a rigorous steepest descent analysis of the Tracy-Widom formula for the asymmetric simple exclusion with anti-shock initial data, which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit.

##### Diffusion in a periodic Lorentz gas with narrow tunnels (P. Balint, N. Chernov, and D. Dolgopyat)

In a periodic Lorentz gas a particle moves bouncing off a regular array of fixed convex obstacles (scatterers), like in a pinball machine. When the horizon is finite, one observes a classical diffusion law. When the obstacles are so large that the tunnels between them become narrow (of width $\epsilon \to 0$) then the diffusion matrix scales with $\epsilon$. In the limit, when $\epsilon=0$, the particle is trapped in a compact region with cusps in the boundary. In that case the system ceases to be uniformly hyperbolic and develops anomalies. Correlations decay slowly, as $1/n$, and the classical central limit theorem fails. Instead, a non-classical limit law holds, with a scaling factor of $\sqrt{n\log n}$ replacing the standard $\sqrt{n}$.

##### Ergodic properties of infinite extensions of area-preserving flows

We consider infinite volume preserving flows that are obtained as extensions of flows on surfaces. Consider a smooth area-preserving flow on a surface $S$ given by a vector field $X$ and consider a real valued function $f$ on $X$. The (skew-product) extension of the flow on $S$ given by $f$ is the flow on $S x R$ given by the solutions to the differential equations $dx/dt=X$, $dy/dt=f$, where $(x,y)$ are coordinates on $S x R$ and $R$ is the real line. While the ergodic properties of surface flows that preserve a smooth area form are now well understood (as we will summarize), very little is known for these infinite measure preserving extensions, which were previously studied only when S is a torus.

##### Bifurcations of solutions of the 2-dimensional Navier-Stokes system

I will explain recent joint work with Sinai on the bifurcations of solutions to the 2-dimensional Navier-Stokes system.

##### Local semicircle law at the edge of spectrum for Gaussian $\beta$-ensembles

We prove, via the moments method, a local semicircle law for the Gaussian $\beta$-ensembles at the almost optimal scaling of $n^{-2/3 +\epsilon}$ at the edge of the spectrum for all $\beta \geq 1$.

##### On the Vershik-Kerov Conjecture Concerning the Shannon-McMillan-Breiman Theorem for the Plancherel Family of Measures on the Space of Young Diagrams

Vershik and Kerov conjectured in 1985 that dimensions of irreducible representations of finite symmetric groups, after appropriate normalization, converge to a constant with respect to the Plancherel family of measures on the space of Young diagrams. The statement of the Vershik-Kerov conjecture can be seen as an analogue of the Shannon-McMillan-Breiman Theorem for the non-stationary Markov process of the growth of a Young diagram. The limiting constant is then interpreted as the entropy of the Plancherel measure. The main result of the paper is the proof of the Vershik-Kerov conjecture. The argument is based on the methods of Borodin, Okounkov and Olshanski. The talk is based on the preprint arXiv:1001.4275

##### Polynomials in the Poisson scaling regime and gaps in $n^{1/3}\mod 1$

I will show that a quadratic polynomial exhibits "random" behavior on the circle in the Poisson scaling regime. Specifically I will prove that second moments are Poissonian and motivate why other moments could be Poissonian. An essential feature of the polynomial under consideration is that it has a sufficiently "random" constant term. In the corresponding exponential sum the main term will not come from "major arcs" but from other points. Finally, I will relate this problem to gaps in $n^{1/3}\mod 1$ and show that Poisson moments (only two moments known at this time) for a specific quadratic polynomial imply exponential distribution for gaps.

##### Relative Entropy Indicators and applications to Authorship Attribution

Estimating the entropy of an unknown (ergodic, stationary) stochastic source with unknown memory, only relying on finite samples, poses interesting and challenging mathematical questions. In this talk we aim first to review some recent and less recent relative entropy indicators, such as the zipper's based methods. Then we will recall the main definitions and results on non-sequential recursive pair substitutions and their use as relative entropy indicators. Finally, we will discuss some recent applications to authorship attribution of literary texts.