Ergodic Theory and Statistical Mechanics Seminar AY2012-2013 (See current year)

Thursdays 2:00-3:30pm, Room 601, Fine Hall, Princeton University.

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 Date: May 2nd 2013 Speaker: Evgeni Dimitrov (Princeton University) Title: On some problems in Diophantine approximation Abstract: The purpose of my talk will be to present some new results for the Littlewood and Mixed Littlewood conjectures. In the first part of the discussion I will give an expository overview of what is currently known about the two conjectures and present in more detail a particular problem connected to them. In the second part I will give an overview of the arguments that go into proving the problem for the mixed case. Date: April 25th 2013 Speaker: Gabriel Koch (University of Sussex) Title: Navier-Stokes regularity criteria Abstract: 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. Date: Friday, April 19th 2013 Speaker: Francesco Cellarosi (University of Illinois, Urbana-Champaign) Title: Ergodic properties of m-free integers in number fields Abstract: 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. Date: April 18th 2013 Speaker: Alexander Bufetov (LATP-CNRS, Marseille; Steklov Institute, Moscow; Rice University, Houston) Title: Infinite Determinantal Measures Abstract: 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. Date: April 11th 2013 Speaker: Tatyana Shcherbyna (Institute for Advanced Study) Title: Universality of the second mixed moment of the characteristic polynomials of the 1D Gaussian band matrices Abstract: 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. Date: April 4th 2013 Cancelled Speaker: Jürg Fröhlich (Institute for Advanced Study) Title: The problem of dynamics in quantum theory Abstract: To begin with, a rather abstract algebraic approach to the formulation of dynamical models of physical systems is described. This approach is general enough to encompass classical and quantum-mechanical models. The main features of classical models are briefly recapitulated. It is then explained in which way quantum-mechanical models are fundamentally different from classical ones. The key properties of quantum dynamics distinguishing it from classical dynamics are related to the concepts of entanglement generation and of "intrinsic information loss". After explaining these concepts and some general results about them, some concrete quantum systems are described, and it is explained what properties their dynamics have. In particular, models of particles coupled to environments with infinitely many degrees of freedom (e.g., random potential landscapes, quantum fields or quantum gases) are presented that exhibit the phenomena of localization, diffusive (quantum Brownian) motion, motion with friction and ballistic motion. To conclude, Mott's problem of the generation of particle tracks is discussed briefly. Date: March 28th 2013 Speaker: Senya Shlosman (Center for Theoretical Physics, France) Title: $N^{1/3}$ scaling in the 2D Ising model and the Airy diffusion. Abstract: 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. Date: March 7th 2013 Speakers: Ivan Christov and Howard Stone (Princeton University) Title: Diffusion in flows of granular materials Abstract: Flowing granular materials are an example of a heterogeneous complex system away from equilibrium. As a result, their dynamics are still poorly understood. One canonical example is granular flow in a slowly-rotating container. Granular materials do not perform Brownian motion, nevertheless diffusion is observed in such systems because agitation (flow) causes inelastic collisions between particles. It has been suggested that axial diffusion of granular matter in a rotating drum might be "anomalous" in the sense that the mean squared displacement of particles follows a power law in time with exponent less than unity. Further numerical and experimental studies have been unable to definitively confirm or disprove this observation. In this talk, we will first review the theory of fractional parabolic PDEs governing anomalous diffusion and discuss their physical origin. Next, we will show that such a "paradox" can be resolved using the theory of self-similar intermediate asymptotics of (nonlinear) parabolic PDEs, without the need to appeal to nonlocal effects such as those represented by fractional derivatives. Specifically, we will derive an analytical expression for the instantaneous scaling exponent of a macroscopic concentration profile. Then, by incorporating concentration-dependent diffusivity in the model, we will show the existence of a crossover from an anomalous scaling (consistent with experimental observations) to a normal diffusive scaling at very long times. Date: February 28th 2013 Speaker: David Ruelle (Institut Des Hautes Études Scientifiques) Title: Lee-Yang Zeros, And Applications To Graph-Counting Polynomials. Abstract: 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. Date: February 14th 2013 Speaker: Nalini Anantharaman (Institute for Advanced Study) Title: Entropy and the localization of eigenfunctions on negatively curved manifolds - I Abstract: 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 I : I will state the result, introduce some notions of microlocal analysis, recall the definition of entropy and state a technical estimate which is the core of the proof. Date: February 21st 2013 Speaker: Nalini Anantharaman (Institute for Advanced Study) Title: Entropy and the localization of eigenfunctions on negatively curved manifolds - II Abstract: 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. Date: Friday, November 30th 2012 Speaker: Philip Isett (Princeton University) Title: Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time II Abstract: 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. Date: November 29th 2012 Speaker: Philip Isett (Princeton University) Title: Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time I Abstract: 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. Date: November 15th 2012 Speaker: Evgeni Dimitrov (Princeton Univeristy) Title: On Littlewood’s conjecture in Diophantine approximation Abstract: The purpose of my talk will be to present some of the results that have been established for Littlewood’s conjecure. 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.” Date: Tuesday, November 13th 2012 Room: Fine 801 Speaker: Ilya Goldsheid (Queen Mary, University of London) Title: Traps and Random Walks in Random Environments on a Strip. Date: November 8th 2012 Speaker: Domokos Szász (Budapest University of Technology) Title: On Dettmann's 'Horizon' Conjectures Abstract: 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). According to Dettmann's first conjecture the probability that the so determined free flight (until the first hitting of a ball) is larger than $t \gg 1$ is $\sim\frac{C}{t}$ where $C$ is explicitly given by the geometry of the model. In its simplest form, Dettmann's second conjecture is related to the previous case with tangent balls (of radii $1/2$). The conjectures are established in a more general setup: for $\mathcal{L}$-periodic configuration of complex bodies with $\mathcal{L}$ being a non-degenerate lattice. These questions are related to Pólya's visibility problem (1918), to the results of Bourgain-Golse (1998-) and of Marklof-Stroembergsson (2010-). The results, joint with P. Nándori and T. Varjú, also provide the asymptotic covariance of the periodic Lorentz process assuming it has a limit in the super-diffusively scaling, a fact if $d=2$ and the horizon is infinite. Date: October 25th 2012 Speaker: Leonid Petrov (Northeastern University) Title: Random lozenge tilings of polygons and their asymptotic behavior Abstract: 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). Bulk asymptotic behavior allows to reconstruct the limit shapes of random stepped surfaces obtained by Cohn, Propp, Kenyon, and Okounkov. We also prove a conjecture of Kenyon (2004) that the large-scale asymptotics of random tilings are asymptotically governed by the Gaussian Free Field. As a particular case, all our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon). Date: October 18th 2012 Speaker: Percy Wong (Princeton University) Title: Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices Abstract: 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. We also give a variance bound which implies the CLT for test functions in the Sobolev space $H^{1+\epsilon}$ and $C^{1-\epsilon}$ for general Wigner matrices satisfying moment conditions. Previous results on the CLT impose the existence and continuity of at least one classical derivative. Joint work with P. Sosoe. Date: Tuesday, October 16th 2012 Speaker: Michael Yampolsky (Univeristy of Toronto) Title: The fixed point of parabolic renormalization Abstract: 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. Date: October 11th 2012 Speaker: Jack Hanson (Princeton Univeristy) Title: Geodesics in 2D First-Passage Percolation Abstract: 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. Date: October 4th 2012 Speaker: Edriss Titi (Univeristy of California, Irvine) Title: On the Loss of Regularity for the Three-Dimensional Euler Equations Abstract: 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. This is very different from what has been proven for the two-dimensional Kelvin-Helmholtz problem where a minimal regularity implies the real analyticity of the interface. Eventually, we use this shear flow to provide explicit examples of non-regular solutions of the three-dimensional Euler equations that conserve the energy, an issue which is related to the Onsager conjecture. In addition, we will use this shear flow to provide a nontrivial example for the use of vanishing viscosity limit, of the Navier-Stokes solutions, as a selection principle for uniqueness of weak solutions of the 3D Euler equations. Date: September 27th 2012 Speaker: Thomas Beck (Princeton University) Title: Duchon-Robert Solutions for a Two-Fluid Interface Abstract: 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. Last update: 2013-05-22