# Seminars & Events for Ergodic Theory & Statistical Mechanics

##### Limiting Distribution of Large Frobenius Numbers

##### The decay of Fourier modes of solutions of 2-D Navier-Stokes system

Also:

Numerical results related to the talk by D. Li

presented by Nikolai I. Chernov, University of Alabama

##### Limit lognormal process, Selberg integral as Mellin transform, and intermittency differentiation.

The limit lognormal process is a multifractal stochastic process with the remarkable property that its positive integral moments are given by the celebrated Selberg integral. We will give an overview of the limit lognormal construction followed by a summary of our results on functional Feynman-Kac equations and resulting intermittency expansions that govern its distribution. The talk will focus on the intermittency expansion for the Mellin transform. This expansion recovers Selberg’s formula for the positive integral moments and gives a novel product formula for the negative ones. By summing it in general using a moment constant method, we obtain an extension of Selberg’s finite product to the Mellin transform of a probability distribution in the form of an infinite product of ratios of gamma functions in the complex plane.

##### What makes the ergodic theory of Markov chains in infinite dimensions different (and difficult)?

I will discuss how Markov chains in infinite dimensions generically have typically have properties which make their ergodic theory difficult. Such properties are very pathological in finite dimensions, but in some sense generic in infinite dimensions. I will draw examples from stochastically forced PDEs and stochastic delay equations. We will see that in infinite dimensions, a typical system acts much more like an hypo-elliptic diffusion then an elliptic one. I will also discuss the existence of spectral gaps as well as the uniqueness of invariant measures. If time permits I will discuss an extension of Hormander's "sum of squares theorem" to infinite dimensions.

##### Hénon Renormalization

The geometry of strongly dissipative infinite renormalizable Hénon maps of period doubling type is surprisingly different from its one-dimensional counterpart. There are universal geometrical properties. However, the Cantor attractor is not geometrically rigid. Typically, it doesn't have bounded geometry. The average Jacobian is a topological invariant of the global attractor. Although the geometry of the Cantor attractor can be deformed by changing the average Jacobian, the geometry is universal in a distributional sense.

##### Random walks with memory and statistical mechanics

This talk will review some results and conjectures about history dependent random walks. For example, edge reinforced random walk (ERRW) is a random walk which prefers to visit edges it has visited in the past. Diaconis showed that ERRW can be expressed as a random walk in a random environment. This environment is highly correlated and is described in terms of statistical mechanics. Phase transitions for closely related models are believed to occur in three dimensions. One phase corresponds to diffusion and the other phase to localization. This talk is based work of Merkl and Rolles on ERRW and my recent preprint with Disertori and Zirnbauer on a hyperbolic sigma model.

##### Local limit theorems in ergodic theory

We use Stone's version of a local limit theorem from 1969: Let $(X,{\cal F},T,m)$ be a measure preserving dynamical system. A measurable function $f:X\to \mathbb R$ satisfies a local limit theorem, if there are constants $A_n$ and $B_n\to\infty$ such that $$ B_nm( f+f\circ T+...+f\circ T^{n-1} \in x_n+I) \to g(x)|I|,$$ where $(x_n-A_n)/B_n \to x$ and where $g$ is the density of some stable distribution. An analogous definition applies in the lattice case.

##### Large deviations of the current and phase transitions

Using the framework of the hydrodynamic limits, we will discuss the large deviations of a particle current through a diffusive system. The deviations can lead to dynamical phase transitions. In the case of asymmetric dynamics we will explain how the large deviation functional of the current provides a physical interpretation to the non-entropic solutions of Burgers equation.

##### On mixing properties of locally Hamiltonian flows on surfaces

We consider area-preserving flows on surfaces which are locally given by smooth Hamiltonians. It turns out that the presence or absence of mixing depends on the type of fixed points. We proved in our PhD thesis that the presence of centers is generically enough to create mixing. Recently we showed that if such flows have only saddles, they are generically not mixing, but weakly mixing. The results use the flows representation as suspensions over interval exchange transformations and the study of deviations of Birkhoff averages over interval exchanges.

##### Existence/Non-existence of limiting distributions for horocycle flows on compact surfaces of constant negative curvature

A few years ago we have proved in collaboration with L. Flaminio that some non-trivial limit distributions for the horocycle flow must have compact support. In this talk we will refine that result and describe an existence/non-existence result for limiting distributions. In fact it turns out that whether limiting distributions exist or not depends on the geometry of the surface (via the eigenvalues of the Laplace operator) and on the observable under consideration. The main new idea is to express the precise results on the asymptotics of ergodic averages for the horocycle flow in terms of a dynamically defined cocycle which has the correct scaling property under the dynamics of the geodesic flow. Such cocycles are closely related to the invariant distributions of Flaminio-Forni and are analogous to the coycles constructed by A.

##### An explicit approach to the control of Lyapunov exponents

I shall discuss a new approach to the proof the exponential growth of products of random matrices. The classical Furstenberg's analysis relies on properties of infinite-dimensional unitary representations. The method I am going to discuss uses finite-dimensional representations and allows one to have a more explicit control over Lyapunov exponents.

##### Local entropy and projections of dynamically defined fractals

If a closed subset $X$ of the plane is projected orthogonally onto a line, then the Hausdorff dimension of the image is no larger than the dimension of $X$ (since the projection is Lipschitz), and also no larger than $1$ (since it is a subset of a line). A classical theorem of Marstrand says that for any such $X$, the projection onto almost every line has the maximal possible dimension given these constraints, i.e. is equal to $min(1,dim(X))$. In general, there can be uncountably many exceptional directions.

##### Lee-Yang zeros for the Diamond Hierarchical Lattice and 2D rational dynamics

In a classical work of 1950's, Lee and Yang proved that zeros of the partition functions of the Ising models on graphs always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for a special "Diamond Hierarchical Lattice." In this case, it can be described in terms of the dynamics of an explicit rational map in two variables. We prove partial hyperbolicity of this map on an invariant cylinder, and derive from it that the Lee-Yang zeros are organized asymptotically in a transverse measure for the central foliation. From the global complex point of view, the zero distributions get interpreted as slices of the Green (1,1)-current on the projective space. It is a joint work with Pavel Bleher and Roland Roeder.