# Seminars & Events for Ergodic Theory & Statistical Mechanics

##### A modification of the moment method and stochastically evolving partitions at the edge

This talk is about a modification of the moment method applied to extract limiting distributions of the first, second, and so on rows of randomly evolving partitions.

We will proceed slowly, first by describing the usual moment method together with a modification using orthogonal polynomials. Here, the central limit theorem and Wigner's semicircle law will be presented as simple examples of how the passage to the modified moments works out.

We then give a brief survey of recent developments in random matrix theory regarding universality in the fluctuations of extreme eigenvalues of random matrix ensembles.

##### Supersymmetric approach in the random matrix theory

In this talk I will give a brief outline of the Grassmann integration technique (which is also called the supersymmetry approach) and its application to the rigorous study of main spectral characteristics of random matrices. I am going to start from a simplest case of GUE matrices, and then to show the relation between characteristic polynomials of random band matrices and some questions about the classical Heisenberg model.

##### Ergodic measures for a class of subshifts

We will consider minimal subshifts with complexity such that the difference from n to n+1 is constant for all large n and impose one more condition (which we call the Regular Bispecial Condition). The shifts that arise naturally from interval exchange transformations belong to this class. A minimal interval exchange transformation on d intervals has at most d/2 ergodic probability measures. This well-known result was due to Katok and later Veech using symplectic arguments. We work to show this bound by combinatorial means on our more general class of subshifts. This is ongoing work with Michael Damron.

##### Shuffling large decks of cards and the Bernoulli-Laplace urn model

In boardgames, in Casino games with multiple decks and in cryptography, one is sometimes faced with the practical problem: how can a human (as opposed to the computer) shuffle a big deck of cards. One natural procedure (used by casino’s) is to break the deck into several reasonable size piles, shuffle each throughly, assemble, do some simple deterministic thing (like a cut) and repeat. G. White and I introduce variations of the classical Bernoulli-Laplace urn model (the first Markov Chain!) involving swaps of big groups of balls. Now, a coupling argument and spherical function theory allow the original problem to be solved.

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##### Stochastic 3D Navier-Stokes Equations with Uniformly Large Initial Vorticity

##### Strong stationary times for small shuffles

Consider shuffling a deck of cards by at each step, choosing two random cards and either swapping them or not. This random walk is well understood. A natural generalisation is to choose three (or more) cards at each step, and shuffle them amongst themselves. I will show how one may construct a strong stationary time for such a random walk, giving an asymptotic upper bound on the mixing time. The resulting bounds will be similar to conjectures of Roichman.

##### A Central Limit Theorem for a $\B$-free dynamical system

For a set $\mathcal{B} \subset \mathbb{N} \setminus \{1\}$, let $\mathcal{B}$-free integers be the set of integers that are not divisible by any element of $\mathcal{B}$ and let $X^{\mathcal{B}} \subset \{0,1\}^{\mathbb{Z}}$ be the closure of the orbit of the indicator of $\mathcal{B}$--free integers under the left shift $T$. One can equip $\left(X^{\mathcal{B}},T\right)$ with the $T$--invariant measure which produces the correlations of $\mathcal{B}$-free integers. For an infinite, coprime $\mathcal{B}$ with $\sum_{b \in \mathcal{B}} 1/b <\infty$, the resulting dynamical system is measurably isomorphic to an ergodic shift on a compact abelian group (el Abdalaoui, Lema\'nczyk \& de la Rue).

##### Quasi-periodic solutions to nonlinear PDE

We discuss time quasi-periodic solutions to nonlinear Schroedinger (NLS) and nonlinear wave equations (NLW) on the torus in arbitrary dimensions. The latter is hyperbolic and uses additionally, a Diophantine property of algebraic numbers. We mention also a work in progress, on space-time quasi-periodic solutions to non-integrable NLS, whose analysis rests on semi-algebraic geometry.

##### Possible lattice distortions in the Hubbard model for graphene

**Please note special time and location.**

We consider the Hubbard model on the honeycomb lattice as a model for graphene and ask the question of possible lattice distortions. The answer is that in the thermodynamic limit only periodic, reflection-symmetric distortions are allowed and these have at most six atoms per unit cell as compared to two atoms for the undistorted lattice. We sketch the proof, which is based on the technique of reflection positivity.

The talk is based on a joint work with E. Lieb.

##### Families of mild mixing interval exchange transformations

Almost every interval exchange transformation is rigid. In this talk I will describe recent work showing that for an infinite class of permutations the set of mild mixing interval exchange transformations has full Hausdorff dimension.

##### A geometric approach for constructing SRB measures

I will describe a unified approach for constructing SRB measures which is pure geometrical and does not use any symbolic model of the system. This approach originated in the work of Sinai. I will first outline a construction of SRB measures in the classical case of uniformly hyperbolic attractors and then consider the most general case of non-uniformly hyperbolic chaotic attractors. This is a joint work with Vaughn Climenhaga and Dima Dolgopyat.

##### A few fairy math tales

The lecture consists of several mini-talks with just definitions, motivations, some ideas of proofs, and open problems. I will discuss some or all of the following topics. 1. “A survival guide for feeble fish”. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water flow. This is related to homogenization of G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov.2. How well can we approximate an (unbounded) space by a metric graph whose parameters (degree of vertices, length of edges, density of vertices etc) are uniformly bounded? We want to control the ADDITIVE error.

##### The ground state construction of bilayer graphene

We consider a model of weakly-interacting electrons in bilayer graphene. Bilayer graphene is a 2-dimensional crystal consisting of two layers of carbon atoms in a hexagonal lattice. Our main result is an expression of the free energy and two-point Schwinger function as convergent power series in the interaction strength. In this talk, I discuss the properties of the non-interacting model, and exhibit three energy regimes in which the energy bands are qualitatively different. I then sketch how this decomposition may be used to carry out the renormalization group analysis used to prove our main result. Joint work with Alessandro Giuliani.

##### Error in Central Limit Theorem for i.i.d. random variables with atomic distribution

We consider sums of independent identically distributed random variables whose distributions have d atoms. Such distributions never admit an Edgeworth expansion of order d-1 but we show that for almost all

parameters the Edgeworth expansion of order d-2 is valid and the error of the order d-1 Edgeworth expansion is typically of order n^{-(d-1)/2}. This is a joint work with Kasum Fernando.

##### General relationship among evolvability, fluctuations and robustness: Selection of/by/for dynamical systems

**PLEASE NOTE DIFFERENT TIME (11:00 A.M.) AND LOCATION. **In biological evolution under given fitness, selection process is based on phenotypes (i.e., a set of state variables such as concentrations of chemicals in a cell), which are shaped by dynamical systems whose rule are given by genes which change through evolution. In other words, dynamical systems are selected as a result of dynamical systems. From simulations of such evolving dynamical systems as well as from bacterial evolution experiments, we first have uncovered evolutionary fluctuation-response relationship, i.e., proportionality between phenotypic variances by noise in dynamics and by genetic variation.

##### Large holes in particle systems : forbidden regions, large deviations and potential theory

In particles systems, a "hole'' of size R is defined to be a ball of radius R that is devoid of particles. The study of how the probability of having such a hole decays to 0 (as R -> infty) is an important and well-studied question in particle systems. In this talk, we ask what causes a large hole to appear? In other words, conditioned on having a large hole, how does the configuration of particles outside the hole look like? Surprisingly, very little is understood about this question, except in the very special case of Gaussian random matrix ensembles, where there is an accumulation of particles at the edge of the hole, and equilibrium intensity beyond. We study this question in the context of zeros of Gaussian random polynomials, and provide a complete description of the intensity profile of the outside particles.

##### Dynamics on Riemann surfaces and extensions in higher dimensions

Dynamical systems in low dimensions, such as interval exchanges or flows on surfaces, come in natural families and their moduli spaces are objects of intrinsic interest. Riemann surfaces and their geometry play a key role, and after introducing the basics I will give an overview of recent results in this area. I will then discuss extensions of these concepts to higher dimensions, where the geometry of K3 surfaces comes in. Generalizing counts of closed billiard trajectories in rational-angled polygons, I will explain how to count a higher-dimensional analogue on K3s. The necessary background will be provided.

##### On rank and isomorphism of von Neumann special flows

**Please note special time and location. **A von Neumann flow is a special flow over an irrational rotation of the circle and under a piecewise smooth roof function with a non-zero sum of jumps. Such flows appear naturally as special representations of Hamiltonian flows on the torus with critical points. We consider the class of von Neumann flows with one discontinuity. I will show that any such flow has infinite rank and that the absolute value of the jump of the roof function is a measure theoretic invariant. The main ingredient in the proofs is a Ranter type property of parabolic divergence of orbits of two nearby points in the flow direction. Joint work with Adam Kanigowski.

##### Invariant measure for random walks on ergodic environments on a strip

This is a joint work with D. Dolgopyat. We consider a random walk (RW) in ergodic random environment (RE) on a strip in the 'environment viewed from the particle' setting. It is well known that in this setting the RW is a Markov chain on the set of environments. This approach to RWRE was introduced by S. Kozlov (1978, 1985) as well as Papanicolaou and Varadhan (1982) and the related fundamental question in this context is: Does this Markov chain have an invariant measure with a density with respect to the measure on the set of environments? It turns out that in the case of the walks on a strip in ergodic RE it is possible to derive the necessary and sufficient conditions for the existence of the density in all regimes - transient and recurrent. We also answer some of the questions asked by Ya.

##### Effective Equidistribution and Hecke Operators in Dynamical Systems

Hecke Operators are ubiquitous in the theory of automorphic forms. We present a simple construction of averaging operators on state spaces of measurable dynamical systems. This is a common generalization of Hecke operators from number theory and Markov shifts. It is also closely related to the Laplacian on a Riemannian manifold. Our first objective is showing an effective ergodic theorem with an exponential rate — large deviations — using a norm gap for the averaging operator. I will present a general criterion for states spaces of dynamical systems which implies a relatively sharp large deviations result. A large class of such systems arises from S-arithmetic quotients of reductive groups. This part builds upon the work of Kahale and Ellenberg, Michel and Venkatesh.