# Seminars & Events for Ergodic Theory & Statistical Mechanics

##### Duchon-Robert Solutions for a Two-Fluid Interface

Duchon and Robert constructed a class of global vortex sheet solutions to the Euler equations, where the vorticity is concentrated on an analytic curve for all positive time. In this talk, I will first discuss the main ideas behind their construction and general properties of vortex sheet solutions. I will then describe an extension of their work to the case of a two-fluid interface. This is joint work with Philippe Sosoe and Percy Wong.

##### On the Loss of Regularity for the Three-Dimensional Euler Equations

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the immediate loss of smoothness and ill-posedness of solutions of the three-dimensional Euler equations, for initial data that do not belong to $C^{1,\alpha}$. Moreover, we show by means of this shear flow example the existence of weak solutions for the three-dimensional Euler equations with vorticity that is having a nontrivial density concentrated on non-smooth surface.

##### Geodesics in 2D First-Passage Percolation

I will discuss geodesics in first-passage percolation, a model for fluid flow in a random medium. C. Newman and collaborators have studied questions related to existence and coalescence of infinite geodesics under strong assumptions. I will explain recent results with Michael Damron which develop a framework for addressing these questions; this framework allows us to prove forms of Newman's results under minimal assumptions.

##### The fixed point of parabolic renormalization

**Please note special day (Tuesday). **I will present our joint work with O. Lanford on the parabolic renormalization operator acting on the space of simple parabolic analytic germs. We have constructed a renormalization-invariant class of analytic maps with a maximal domain of analyticity and rigid covering properties.This class contains the Inou-Shishikura renormalization fixed point. I will also discuss an approach to numerical computation of the renormalization fixed point map based on an asymptotic expansion for the Fatou coordinate, which is resurgent in the sense of J. Ecalle.

##### Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices

We show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large $n$ for some classes of test functions less regular than Lipschitz functions. This observation is suggested by the limiting form of the variance (which has previously been computed explicitly), but it does not seem to appear in the literature. We combine this fact with comparison techniques following Tao-Vu and Erd\"os, Yau, et al. and a Littlewood-Paley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the H\"older class $C^{1/2+\epsilon}$ in the case of matrices of Gaussian convolution type.

##### Random lozenge tilings of polygons and their asymptotic behavior

I will discuss the model of uniformly random tilings of polygons drawn on the triangular lattice by lozenges of three types (equivalent formulations: dimer models on the honeycomb lattice, or random 3-dimensional stepped surfaces). Asymptotic questions about these tilings (when the polygon is fixed and the mesh of the lattice goes to zero) have received a significant attention over the past years. Using a new formula for the determinantal correlation kernel of the model, we manage to establish the conjectural local asymptotics of random tilings in the bulk (leading to ergodic translation invariant Gibbs measures on tilings of the whole plane), and the predicted behavior of interfaces between so-called liquid and frozen phases (leading to the Airy process).

##### On Dettmann's 'Horizon' Conjectures

In the simplest case consider a $\mathbb{Z}^d$-periodic ($d \geq 3$) arrangement of balls of radii $< 1/2$, and select a random direction and point (outside the balls). CLICK ON THE SEMINAR TITLE FOR THE COMPLETE ABSTRACT.

##### Traps and Random Walks in Random Environments on a Strip

PLEASE NOTE SPECIAL DAY (TUESDAY, NOV. 13)

##### On Littlewood’s conjecture in Diophantine approximation

The purpose of my talk will be to present some of the results that have been established for Littlewood’s conjecture. In the first part of the discussion I will give an expository overview of what is currently known about the conjecture and what are some of the questions that naturally arise from it. In the second part I will present in more detail the seminal work of Andrew Pollington and Sanju Velani from their 2001 paper: “On a problem in simultaneous Diophantine approximation: Littlewood’s conjecture.”

##### Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time I

In connection to the theory of hydrodynamic turbulence, Onsager conjectured that solutions to the incompressible Euler equations with Holder regularity below 1/3 may dissipate energy. Recently, DeLellis and Székelyhidi have adapted the method of convex integration to construct energy-dissipating solutions with regularity up to 1/10. In a two lecture series, we will discuss Onsager’s conjecture, its relation to turbulence, and how one can use convex integration to construct solutions with regularity up to 1/5 which have compact support in time.

##### Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time II

In connection to the theory of hydrodynamic turbulence, Onsager conjectured that solutions to the incompressible Euler equations with Holder regularity below 1/3 may dissipate energy. Recently, DeLellis and Székelyhidi have adapted the method of convex integration to construct energy-dissipating solutions with regularity up to 1/10. In a two lecture series, we will discuss Onsager’s conjecture, its relation to turbulence, and how one can use convex integration to construct solutions with regularity up to 1/5 which have compact support in time.

##### Entropy and the localization of eigenfunctions on negatively curved manifolds - I

We are interested in the behaviour of laplacian eigenfunctions on negatively curved manifolds, in the high frequency limit. The Quantum Unique Ergodicity conjecture predicts that they should become uniformly distributed over phase space, and the Shnirelman theorem states that this is true if we allow ourselves to possibly drop a ``negligible'' family of eigenfunctions. Nonnenmacher and I proved that, in any case, the eigenfunctions must in the high frequency regime have a large Kolmogorov-Sinai entropy : this prevents them, for instance, from concentrating on periodic geodesics. The proof uses notions from ergodic theory (such as entropy) mixed with techniques from linear PDE.

##### Entropy and the localization of eigenfunctions on negatively curved manifolds - II

We are interested in the behaviour of laplacian eigenfunctions on negatively curved manifolds, in the high frequency limit. The Quantum Unique Ergodicity conjecture predicts that they should become uniformly distributed over phase space, and the Shnirelman theorem states that this is true if we allow ourselves to possibly drop a ``negligible'' family of eigenfunctions. Nonnenmacher and I proved that, in any case, the eigenfunctions must in the high frequency regime have a large Kolmogorov-Sinai entropy : this prevents them, for instance, from concentrating on periodic geodesics. The proof uses notions from ergodic theory (such as entropy) mixed with techniques from linear PDE.

Talk II : I will prove the technical estimate and finish the proof of the main theorem.

##### Lee-Yang Zeros, And Applications To Graph-Counting Polynomials

The Lee-Yang circle theorem is the prototype of a class of results giving very precise information on the locus of the zeros of certain families of (arbitrarily high degree)polynomials in one variable. The standard application of these results is to statistical mechanics, but we shall see that there are also results about graph-counting polynomials. The intriguing feature of all these results is that they appear quite inaccessible by more standard methods of study of polynomials.

##### Diffusion in flows of granular materials

Flowing granular materials are an example of a heterogeneous complex system away from equilibrium. As a result, their dynamics are still poorly understood. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### $N^{1/3}$ scaling in the 2D Ising model and the Airy diffusion

I will consider the behavior of the phase separation interface of the 2D Ising model in the vicinity of the wall. Properly scaled, it converges to the diffusion process, with a drift expressed via Airy function. This process has appeared first in Ferrari&Spohn's study of Brownian motion, constrained to stay outside circular barrier. Joint work with D. Ioffe and Y. Velenik.

##### Universality of the second mixed moment of the characteristic polynomials of the 1D Gaussian band matrices

We consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of the 1D Gaussian band matrices. Assuming that the width of the band grows faster than $\sqrt{N}$, where $N$ is a matrix size, we show that this asymptotic behavior in the bulk of the spectrum coincides with those for the Gaussian Unitary or Orthogonal Ensemble.

##### Infinite Determinantal Measures

Infinite determinantal measures introduced in the talk are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal process and a convergent, but not integrable, multiplicative functional. The main result of the talk gives an explicit description for the ergodic decomposition of infinite Pickrell measures on the spaces of infinite complex matrices in terms of infinite determinantal measures obtained by finite-rank perturbations of Bessel point processes. The talk is based on the preprint arXiv:1207.6793.

##### Ergodic properties of m-free integers in number fields

**PLEASE NOTE SPECIAL DATE.** For an arbitrary number field $K/Q$ of degree $d$, we study the n-point correlations for $m$-free integers in the ring $O_K$ and define an associated natural $O_K$-action. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a $Z^d$ action on a compact abelian group. As a corollary, we obtain that this natural action is not weakly mixing and has zero measure-theoretical entropy. The case $K=Q$, was studied by Ya.G. Sinai and myself, and our theorem provides a different proof to a result by P. Sarnak. This is a joint work with I. Vinogradov.

##### Navier-Stokes regularity criteria

We generalize a well-known result due to Escauriaza-Seregin-Sverak by showing that Navier-Stokes solutions cannot develop a singularity if certain scale-invariant spatial Besov norms remain bounded in time. Our main tool is profile decompositions for bounded sequences in Banach spaces, and we follow the general dispersive method of "critical elements" developed by Kenig-Merle. This is joint work with I. Gallagher and F. Planchon, based on previous work with C. Kenig.