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# Timetable (PEPW04)

## Random and other ergodic problems

Monday 22nd June 2015 to Friday 26th June 2015

 09:00 to 09:50 Registration 09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director) 10:00 to 11:00 On non-smooth functions of Wiener-Hopf operators We discuss trace formulae for the operator \begin{equation*} f(PAP) - Pf(A)P, \end{equation*} where $A$ is a pseudo-differential operator on $L^2(\mathbb R^d)$ with a smooth or discontinuous symbol, and $P$ is a multiplication by the indicator of a piece-wise smooth domain in $\mathbb R^d$. The function $f$ is not supposed to be smooth. The obtained formulae generalise results obtained by H. Widom in the 80's. These results are used to study the entanglement entropy of free fermions at positive temperature both in the low and high temperature limits. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 On the two-dimensional random walk in an isotropic random environment This is joint work with Erich Baur (Lyon) and Ofer Zeitouni (Weizmann Institute). We report on work in progress on the standard model of a random walk in random environment in the critical dimension 2. We investigate exit distributions from large sets which are supposed to be essentially the same as those for ordinary random walks. Random walks in random environment are well understood in dimension one, and for small disorder in dimensions above 2, but the two-dimensional case is largely open. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 From the mesoscopic to microscopic scale in random matrix theory Co-authors: Laszlo Erdos (IST), Horng Tzer Yau (Harvard), Jun Yin (Wisconsin Madison) Eugene Wigner has envisioned that the distributions of the eigenvalues of large Gaussian random matrices are new paradigms for universal statistics of large correlated quantum systems. These random matrix eigenvalues statistics supposedly occur together with delocalized eigenstates. I will explain recent developments proving this paradigm for eigenvalues and eigenvectors of random matrices. This is achieved by bootstrap on scales, from mesoscopic to microscopic. Random walks in random environments, homogenization and the coupling method play a key role. INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 On fluctuations of eigenvalues of random band matrices We consider the fluctuation of linear eigenvalue statistics of random band $n$ dimensional matrices whose bandwidth $b$ is assumed to grow with n in such a way that $b/n$ tends to zero. Without any additional assumptions on the growth of b we prove CLT for linear eigenvalue statistics for a rather wide class of test functions. Thus we remove the main technical restriction $n>>b>>n^{1/2}$ of all the papers, in which band matrices were studied before. Moreover, the developed method allows to prove automatically the CLT for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tales etc. INI 1 16:00 to 17:00 Primitive Pythagorean triples and "near" quasicrystals Counting things is a great favorite of children, and mathematicians as well, whatever the things are. In this talk, I discuss an old counting problem on primitive Pythagorean triples in view of modern theory of quasicrystals. INI 1 17:00 to 18:00 Welcome Wine Reception
 10:00 to 11:00 An eigensystem approach to Anderson localization Co-author: Alexander Elgart (Virginia Tech) We introduce a new approach for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model at high disorder. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems, establishing localization of finite volume eigenfunctions with high probability. (Joint work with A. Elgart.) INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 An eigensystem approach to Anderson localization, part II Co-author: Abel Klein (UC Irvine) We introduce a new approach for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model at high disorder. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems, establishing localization of finite volume eigenfunctions with high probability. (Joint work with A. Klein.) INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 J Schenker (Michigan State University)Dissipative transport in the localized regime Co-author: Jürg Fröhlich (ETH) A quantum particle moving in a strongly disordered random environment is known to be subject to Anderson localization, which results in the complete suppression of transport. However, localization can be broken by a small perturbation, such as thermal noise from the environment, resulting in diffusive motion for the particle. I will discuss this phenomenon in two models in which the Schroedinger equation for a particle in the strongly localized regime is perturbed by (1) a time dependent fluctuating random potential and (2) a Lindblad operator incorporating the interaction with a heat bath in the Markov approximation. In each case, it can be proved that diffusive motion results with a strictly positive and finite diffusion constant. Furthermore, the diffusion constant tends continuously to zero at a calculable rate, as the strength of the perturbation is taken to zero. (Partially based on joint work with J. Fröhlich.) INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 Y Suhov (University of Cambridge)New properties of entropy and their consequences Co-authors: Salimeh Yasaei Sekeh (University of Sao Paulo at Sao Carlos, Brazil), Srefan Zohren (University of Oxford, UK) Based on the concept of the weighted entropy (both classical and quantum), I will report a number of new results involving inequalities and convergence in a variety of context. INI 1 16:00 to 17:00 J Imbrie (University of Virginia)Level Spacing for Non-Monotone Anderson Models INI 1
 10:00 to 11:00 Pure point spectrum in the regime of zero Lyapunov exponents Co-author: Anton Gorodetski (UC Irvine) In the study of ergodic Schrodinger operators, a central role is played by the Lyapunov exponent of the associated Schrodinger cocycle. We discuss a construction showing that the regime of zero Lyapunov exponents can contain pure point spectrum. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Invariance of IDS under Darboux transformation and its application The integrated density of states (IDS) is a crucial quantity for studying spectral properties of ergodic Schroedinger opearators. Especially in one dimension it determines most of the spectral properties. On the other hand, in relation to completely integrable systems, Darboux transformation has been investigated from various points of views. In this talk the invariance of IDS under Darboux transformation will be shown, and as a byproduct the invariance of IDS under KdV flow will be remarked. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 Behavior of the spectrum of the periodic Schrodinger operators near the edges of the gaps Co-author: Leonid Parnovski (UCL) It is a common belief that generically all edges of the spectrum of periodic Schrodinger operators are non-degenerate, i.e. are attained by a single band function at finitely many points of quasi-momentum and represent a non-degenerate quadratic minimum or maximum. We present the construction which shows that all degenerate edges of the spectrum can be made non-degenerate under arbitrary small perturbation. The corresponding perturbation is found in the class of potentials with larger (but proportional) periods; thus the final operator is still periodic but the lattice of periods changes. INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 $L_1$-Estimates for Eigenfunctions of the Dirichlet Laplacian Co-authors: Michiel van den Berg (U Bristol), J\"urgen Voigt (TU Dresden) For $d \in {\bf N}$ and $\Omega \ne \emptyset$ an open set in ${\bf R}^d$, we consider the eigenfunctions $\Phi$ of the Dirichlet Laplacian $-\Delta_\Omega$ of $\Omega$. We do {\it not} require $\Omega$ to be of finite volume. % If $\Phi$ is associated with an eigenvalue below the essential spectrum of $-\Delta_\Omega$, we provide estimates for the $L_1$-norm of $\Phi$ in terms of the $L_2$-norm of $\Phi$ and suitable spectral data of $-\Delta_\Omega$. The main idea in obtaining such estimates consists in finding a---sufficiently small---subset $\Omega' \subset \Omega$ where $\Phi$ is localized in the sense that $\Phi$ decays exponentially as one moves away from $\Omega'$. These $L_1$-estimates are then used in the comparison of the heat content of $\Omega$ at time $t>0$ and the heat trace at times $t' > 0$, where a two-sided estimate is established. \vskip.5em This is joint work with Michiel van den Berg (Bristol) and J\"urgen Voigt (Dresden), with improvements by Hendrik Vogt (Dresden). INI 1 19:30 to 22:00 Conference Dinner at Cambridge Union Society hosted by Cambridge Dining Co.
 10:00 to 11:00 Spectral theory of the Schr?dinger operators on fractals Spectral theory of the Schrodinger operators on fractals (Stanislav Molchanov UNC Charlotte)   Spectral properties of the Laplacian on the fractals as well as related topics (random walks on the fractal lattices, Brownian motion on the Sierpinski gasket etc.) are well understood. The next natural step is the analysis of the corresponding Schrodinger operators and not only with random ”ergodic” potentials (Anderson type Hamiltonians) but also with the classical potentials: fast decreasing, increasing or ”periodic” (in an appropriate sense) ones. The talk will present several results in this direction. They include a) Simon – Spencer type theorem (on the absence of a.c. spectrum) and localization theorem for the fractal nested lattices (Sierpinski lattice) b) Homogenization theorem for the random walks with the periodic intensities of the jumps c) Quasi-classical asymptotics and Bargman type estimates for the Schr?dinger operator with the decreasing gasket d) Bohr asymptotic formula in the case of the increasing to infinity potentials e) Random hierarchical operators, density of states and the non-Poissonian spectral statistics Some parts of the talk are based on joint research with my collaborators (Yu. Godin, A. Gordon, E. Ray, L. Zheng). INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 P Hislop (University of Kentucky)Eigenvalue statistics for random Schrodinger operators Certain natural random variables associated with the local eigenvalue statistics for generalized Anderson-type random Schrodinger operators on the lattice and the continuum in the localization region are distributed according to compound Poisson distributions. In the lattice case the Levy measure of the associated distribution has finite support. Other properties of these random variables will be presented. INI 1 12:30 to 13:30 Lunch at Wolfson Court 13:30 to 14:30 Localisation and ageing in the parabolic Anderson model The parabolic Anderson problem is the Cauchy problem for the heat equation on the d-dimensional integer lattice with random potential. It describes the behaviour of branching random walks in a random environment (represented by the potential) and is being actively studied by mathematical physicists. One of the most important situations is when the potential is time-independent and is a collection of independent identically distributed random variables. We discuss the intermittency effect occurring for such potentials and consisting in increasing localisation and randomisation of the solution. We also discuss the ageing behaviour of the model showing that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time. INI 1 14:30 to 15:00 Afternoon Tea 15:00 to 16:00 The invariant measure for random walks on a strip We explain the necessary and sufficient condition for existence of the invariant measure for the Markov chain on the space of ergodic environments in all regimes. INI 1 16:00 to 17:00 Green's function asymptotic behavior near a non-degenerate spectral edge of a periodic operator Co-authors: Minh Kha (Texas A&M University), Andrew Raich (University of Arkansas) Green's function behavior near and at a spectral edge of a periodic operator is one of what was called by M. Birman and T. Suslina "threshold properties." I.e., it depends on the local behavior of the dispersion relation near the edge. The recent results are presented for the case of a non-degenerate spectral edge (which is conjectured to be the generic situation). This is a joint work with Minh Kha (Texas A&M) and Andrew Raich (Univ. of Arkansas) INI 1
 10:00 to 11:00 Recent results and open problems on many-body localization In the most recent decade the topic of many-body localization, understood as the absence of thermalization in interacting quantum many-body systems, has seen strong attention and rapid development in the physics literature. We will survey the relatively small number of mathematically rigorous results on MBL which have been obtained, in particular for disordered oscillator systems and some models of quantum spin chains. Specifically, we will consider manifestations of MBL such as absence of many-body transport, exponential decay of ground and thermal state correlations, as well as area laws for the entanglement of states. We will also mention some of the many open problems in this field. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 F Klopp (Université Pierre et Marie Curie)Interacting quantum particles in a 1 D random background INI 1 12:30 to 13:30 Lunch at Wolfson Court