CLASSICAL AND QUANTUM MOTION IN DISORDERED ENVIRONMENT

A random event in honour of Ilya Goldsheid's 70-th birthday

Queen Mary, University of London, 18-22/12/2017

Monday 18.12.2017

09:00–09:50 Registration [ArtsTwo Foyer]
09:50–10:00 Opening [ArtsTwo]

Morning session (chair: Parnovski)
10:00–10:40 Leonid Pastur (Kharkov): Large block asymptotics of entanglement entropy of free fermions  slides
10:45–11:10 Coffee break [ArtsTwo Foyer]
11:10–11:50 David Khmelnitskii (Cambridge): Tunnel transparency of a disordered system in perpendicular magnetic field
12:00–12:40 Uzy Smilansky (Rehovot): Anderson Localization in the time domain

12:45–15:20 Lunch

Afternoon session (chair: Chulaevsky)
      cancelled: Senya Shlosman (Marseille): A (four-parameter) continuum of extremal states of the Ising model in 3D halfspace
15:20–16:00 Konstantin Khanin (Toronto): On global solutions to the random heat equation
16:05–16:30 Coffee break
16:30–17:10 Dmitry Dolgopyat (Maryland): Limit theorems for circle rotations

17:30–18:30 Reception [SCR Lounge]
18:30–19:30 Ensemble Le Basile: Arithmetic, Geometry, Music, Astronomy: Hearing the Medieval Quadrivium   programme

Tuesday 19.12.2017

Morning session (chair: Boutet de Monvel)
10:00–10:40 Pavel Bleher (Indianapolis): Exact Solution of the Dimer Model on the Triangular Lattice. Monomer-Monomer Correlation Function  slides
10:45–11:10 Coffee break
11:10–11:50 Mariya Shcherbina (Kharkov): Local eigenvalue statistics of 1d random band matrices  slides
12:00–12:40 Leonid Bunimovich (Atlanta, GA): Finite time dynamics

12:45–14:30 Lunch

Afternoon session (chair: Spitzer)
14:30–15:10 Barry Simon (Pasadena, CA): Szegő–Widom asymptotics for Chebyshev polynomials on subsets of R   slides
15:20–16:00 Günter Stolz (Birmingham, AL): Disordered quantum spin chains  slides
16:05–16:30 Coffee break
16:30–17:10 Gian-Michele Graf (Zurich): The bulk-edge correspondence for disordered chiral chains  slides
       cancelled: Frédéric Klopp (Paris): Resonances for large random samples

Abstracts

Pavel Bleher, Exact Solution of the Dimer Model on the Triangular Lattice. Monomer-Monomer Correlation Function

We will discuss an exact solution for the monomer-monomer correlation function in the dimer model on the triangular lattice. This is a joint work with Estelle Basor.

Erwin Bolthausen, On the Thouless – Anderson – Palmer equations for the perceptron

TAP type equations for the perceptron had first been proposed by Marc Mézard in 1988, but there exists no mathematical proof of their validity. We adapt an iterative scheme for the construction of solutions of the equations, originally introduced 2014 for the SK-model, which is somewhat more complicated for the perceptron. We discuss this and the relation of the TAP equations with the Derrida–Gardner formula for the free energy (work in progress).

Leonid Bunimovich, Finite time dynamics

Traditionally in studies of random systems evolution or dynamics of chaotic systems asymptotic in time properties are investigated. I will demonstrate that some reasonable and interesting questions about finite time evolution can also be answered.

Dmitry Dolgopyat, Limit theorems for circle rotations

Kronecker sequences {x+na} where {…} stands for fractional part are among the most regular sequences on could think of. In my talk I review some results dealing with stochastic behavior of Kronecker sequences.

Anna Erschler, Random walks and Isoperimetry of groups

Alex Furman, Simplicity of the Lyapunov spectrum via Boundary theory

Consider products of matrices in G=SL(d,R) that are chosen using some ergodic dynamical system. The Multiplicative Ergodic Theorem (Oseledets) asserts that the asymptotically such products behave as exp(nΛ) where Λ is a fixed diagonal traceless matrix – the Lyapunov spectrum of the system. The spectrum Λ depends on the system in a mysterious way, and is almost never known explicitly. The best understood case is that of random walks, where by the work of Furstenberg, Guivarc'h-Raugi, and Gol'dsheid-Margulis we know that the spectrum is simple (i.e. all values are distinct) provided the random walk is not trapped in a proper algebraic subgroup. Recently, Avila and Viana proved a conjecture of Kontsevich-Zorich that asserts simplicity of the Lyapunov spectrum for another system related to the Teichmuller flow. In the talk we shall describe an approach to proving simplicity of the spectrum based on ideas from boundary theory that were developed to prove rigidity of lattices. Based on joint work with Uri Bader.

Gian-Michele Graf, The bulk-edge correspondence for disordered chiral chains

Random Schrödinger operators in dimension one exhibit localization, as a rule and as we learnt from Ilya Goldsheid. But there are exceptions. An example is provided by the random Su-Schrieffer-Heeger model, which is an alternating chain of sites between which particles hop. There localization occurs at all but possibly one energy, which is is enough to endow the model with topological features. The model will also be placed in the more general context of symmetry protected topological insulators as belonging to the chiral symmetry class AIII. In particular bulk-edge correspondence will be addressed.

Yves Guivarc'h, Spectral gap properties and asymptotic of extreme values for multivariate affine stochastic recursions

We show that large values of general multivariate stochastic recursions follow extreme value properties of classical type:Fréchet's law,exponential law, Sullivan logarithm law,...The proof uses a multiple mixing property, which follows of a spectral gap property for the affine recursion, and the homogeneity at infinity of the corresponding stationary measure.

Xiaoqin Guo, Quantitative homogenization and Harnack inequality for a degenerate nondivergence form random difference operator

In the d-dimensional integer lattice Z^d, d≥2, we consider a nondivergence form difference operator
L_a u(x) = Σ_1≤i≤d a_i(x)[u(x+e_i) + u(x–e_i) – u(x)], where a(x) = diag(a₁(x),..., a_d(x)), x∈Z^d are nonnegative diagonal matrices which are random i.i.d. with a positive expectation. A difficulty in studying this problem is that coefficients are allowed to be zero. In this talk, using random walks in random media and its percolative structure, we will present a Harnack inequality and a quantitative homogenization result for this random operator. Joint work with N.Berger, M.Cohen and J.-D. Deuschel.

Svetlana Jitomirskaya, Lyapunov exponents, small denominators, arithmetic spectral transitions, and universal hierarchical structure of quasiperiodic eigenfunctions

The work of Goldsheid–Molchanov–Pastur gave the first proof of Anderson localization for 1D random Schrodinger operators, a phenomenon that remarkably holds for all couplings. Unlike random, 1D quasiperiodic Schrodinger operators do feature spectral transitions with changes of parameters. By the classical Kotani and Ishii–Pastur theorems, the transitions between absolutely continuous and singular spectrum are governed by vanishing/non-vanishing of the Lyapunov exponent. In the regime of positive Lyapunov exponents there are also more delicate transitions: between localization and singular continuous spectrum, governed by the competition between the Lyapunov growth and the exponential strength of resonances determined by the arithmetics of frequencies and phases. The fact that such transitions could exist has been known since the work of Avron–Simon in the early 80s. A more recent realization is that in many cases this competition resolves in a sharp way and also leads to arithmetic dependence of dimensional and quantum dynamical properties within the regime of singular continuous spectrum.

We will overview several recent sharp results in this direction, both for general analytic one-frequency quasiperiodic operators in the singular continuous regime, and for specific popular models such as the almost Mathieu operator.

The latter provides in particular the first non-artificial model where Lyapunov–Perron non-regularity can be explicitly studied. For a cocycle over a transformation acting on a space X, non-regular points x ∈ X are the ones at which Oseledets multiplicative ergodic theorem does not hold coherently in both directions. They therefore form a measure zero set with respect to any invariant measure on X. Yet, it is precisely the non-regular points that are of interest in the study of Schrodinger cocycles in the positive Lyapunov exponent regime, since for every x, the spectral measures are supported on energies such that x is a non-regular point of the corresponding non-uniformly hyperbolic transfer-matrix cocycle.

We will present exact exponential asymptotics of all eigenfunctions and of corresponding cocycles for the almost Mathieu operators for all parameters in the localization regime . This uncovers a universal structure in their behavior. For the frequency resonances, there is a hierarchy governed by the continued fraction expansion of the frequency, explaining predictions in physics literature. For the phase resonances, we will describe a new phenomenon, not even previously described in physics: a reflective hierarchy, where self-similarity holds upon alternating reflections.

Konstantin Khanin, On global solutions to the random heat equation

David Khmelnitskii, Tunnel transparency of a disordered system in perpendicular magnetic field

Yuri Kifer, Limit theorems for nonconventional arrays

For about 10 years now together with Varadhan and my students I studied limit theorems for nonconventional sums S_N = Σ_{1 ≤ n≤N} F(X(n), X(2n),...,X(ℓn)), where X(n)'s are random variables with weak dependence, which was partially motivated by nonconventional ergodic theorems originated in Furstenberg's ergodic theory proof of Szemerdi's theorem about arithmetic progressions in sets of integers of positive density. Recently, it turned out that various limit theorems of probability theory can also be studied for sums S_N in a more general situation of nonconventional arrays of the form S_N=Σ_{1 ≤ n≤N} F(X(p₁n+q₁N), X(p₂n+q₂N), ..., X(p_ℓn+q_ℓN)). I will talk about strong law of large numbers, central limit theorem and the Poisson limit theorem for such arrays.

Abel Klein, Manifestations of localization in the random XXZ spin chain

Frédéric Klopp, Resonances for large random samples

The talk will be devoted to the description of the resonances generated by a large sample of random material. In one dimension, one obtains a very precise description for the resonances that directly related to the description fo the eigenvlues and localization centers for the full random model. In higher dimension, below a region of localization in the spectrum for the full random model, one computes the asymptotic density of resonances in some exponentially small strip below the real axis.

Elena Kosygina, Homogenization of viscous Hamilton-Jacobi equations with non-convex Hamiltonian: examples and open questions

Homogenization of Hamilton-Jacobi (HJ) equations with non-convex Hamiltonian in random media is a largely open problem. In the last 4 years several classes of examples and counter-examples of non-convex homogenization appeared in the literature. The majority of known examples concern inviscid HJ equations. We shall discuss two classes of viscous HJ equations with non-convex Hamiltonians in one space dimension for which homogenization holds and pose several open questions. The talk is based on joint work with Andrea Davini (Sapienza – Università di Roma) and with Atilla Yilmaz (Koç University and NYU) and Ofer Zeituni (Weizmann Institute and NYU).

Shin'ichi Kotani, KdV equation with ergodic initial data

The KdV equation is a non-linear partial differential equation ∂_tu = 6 u ∂_xu - ∂_x³u which describes a motion of shallow water wave. This equation is well-known as a completely integrable system, since in 1967 its close connection with one dimensional Schrödinger operators was discovered. Since then, this equation has been investigated by two different methods, one is to use the spectral invariants of the associated Schrödinger operators, and the other is to use Fourier analysis and functional analysis. However, the both methods have treated decaying functions or periodic functions as the initial data, except one paper by Tsugawa studying the local solvability of the equation with quasi-periodic initial data.

We are interested in solutions having complicated oscillation such as of almost periodicity or more generally ergodicity. Algebro-geometric quasi-periodic solutions had been known in the early stage of the research. This case is exactly corresponding to the Schrödinger operators with finite gaps and the reflectionless property on the spectrum. In 1994 Egorova had succeeded to obtain limit periodic solutions. Recently in 2016 Damanik–Goldstein got a certain analytic quasi-periodic solutions, but they had to assume smallness of the initial data. The purpose of the present talk is to develop an algebraic method discovered by Sato and his school. In this trial the representation of the tau-functions by the Weyl–Titchmarsh functions is crucial. So far what we could show is the possibility of the construction of the solutions with ergodic initial data satisfying
∫_{[0,∞)\Σ_ac}λⁿdλ< ∞
for any n≥1, where Σ_ac denotes the absolutely continuous spectrum of the Schrödinger operator with potential of ergodic initial datum. This result contains algebro-geometric solutions, Egorova's limit periodic solutions and Damanik–Goldstein's quasi-periodic solutions.

Ensemble Le Basile, Arithmetic, Geometry, Music, Astronomy: Hearing the Medieval Quadrivium

Randall Cook — Viola d'Arco
Kirsty Whatley — Harp
Uri Smilansky — Viola d'Arco

Programme

Stanislav Molchanov, On the spectrum of the Schrödinger operator associated with the Dyson-Vladimirov Laplacian: cases of the fast-decreasing potentials and the Anderson model

Usually Vladimirov's Laplacian -Δ is defined for L² space on the field of p-adic numbers, but it can also be defined on the extension of the Dyson's lattice. In many senses, it is a "super-fractal": all properties of the classical fractals (like Sierpinski gasket) are presented here in the strongest form: self-similarity, compactly supported Eigen function, large gaps in the spectrum etc.) The talk will present several results on the spectrum of the corresponding Schrödinger operators. First, we will discuss the Cwikel – Lieb –Rozenblum estimates for the number of the negative eigenvalues and also the number of eigenvalues in the gaps (In both cases: recurrent or transient "Brownian motion" with the generator Δ). Secondly, we will present the localization theorem (in the case of unbounded Laplacian) for the appropriate class of random potentials.

Leonid Pastur, Large block asymptotics of entanglement entropy of free fermions

We begin with a brief overview of studies of large block behavior of entanglement entropy of free fermions and related spin chains emphasizing links with the Szegő-type theorems. We then discuss two new results of the field. The first is on the absence of selfaveraging property for the entanglement entropy in one dimension. The second treats finite size effects in the large block behavior of entanglement entropy of free fermions. The effects are due to the possibility to vary the relative order of magnitude of two basic length scales of the problem: the size of the whole system and the size of the block.

Christophe Sabot, Reinforced random walks, random Schrödinger operators and hitting times of Brownian motions

The Vertex Reinforced Jump Process is a continuous time self-interacting process closely related to the Edge Reinforced Random Walk. Its behaviour can be related to the spectral properties at ground state of a random Schrödinger operator with a specific 1-dependent potential. We will review some of the recent results on this topic and also explain how the classical fractional moment method can be used to give a short proof of the localization result of Disertori and Spencer. Finally, we will give a glimpse of new relations with hitting times of a familly of correlated Brownian motions, generalizing the classical relation between 1-D drifted Brownian motion and the inverse Gaussian law.

Mariya Shcherbina, Local eigenvalue statistics of 1d random band matrices

We discuss an application of the transfer operator approach to the analysis of the different spectral characteristics of 1d random band matrices (correlation functions of characteristic polynomials, density of states, spectral correlation functions). We show that when the band width W crosses the threshold W = N^½, the model has a kind of phase transition (crossover), whose nature can be explained by the spectral properties of the transfer operator.

Senya Shlosman, A (four-parameter) continuum of extremal states of the Ising model in 3D halfspace.

Based on joint work with Abraham and Newman, https://arxiv.org/abs/1710.05411

Barry Simon, Szegő–Widom asymptotics for Chebyshev polynomials on subsets of R

Chebyshev polynomials for a compact subset e ⊂ R are defined to be the monic polynomials with minimal ||·||_∞ over e. In 1969, Widom made a conjecture about the asymptotics of these polynomials when e was a finite gap set. We prove this conjecture and extend it also to those infinite gap sets which obey a Parreau–Widom and a Direct Cauchy Theory condition. This talk will begin with a generalities about Chebyshev Polynomials. This is joint work with Jacob Christiansen and Maxim Zinchenko and partly with Peter Yuditskii.

Uzy Smilansky, Anderson Localization in the time domain

Günter Stolz, Disordered quantum spin chains

Numerical work has shown that some models of disordered quantum spin chains exhibit a many-body localization transition. Among the indications of this transition from the localized to the delocalized regime are small or large entanglement of eigenstates, short range versus long range correlations, as well as vanishing versus non-vanishing of the group velocity (expressed in terms of Lieb-Robinson bounds). One of the models where a transition is expected is the XXZ chain in disordered exterior field. We report on joint work with Alexander Elgart and Abel Klein where the Ising phase of this model is studied. This model exhibits droplet formation at low energies and it can be shown that the droplets fully localize in the presence of disorder. Physically, this is interpreted as zero temperature localization. Further results on dynamical localization for this model will be discussed in Abel Klein's talk.

Alain-Sol Sznitman, On porous interfaces and disconnection

In this talk I will present some recent results obtained in collaboration with Maximilian Nitzschner concerning uniform estimates for the absorption by porous interfaces surrounding a compact set of Brownian motion starting in this compact set. I will also discuss some applications to large deviation upper bounds for certain disconnection problems involving coarse graining.

Ofer Zeitouni, Experiments and theorems with noise perturbations of non-normal matrices

Consider an n-dimensional upper triangular matrix A, with empirical measure of eigenvalues n^-1 Σ δ_{A_ii}. What happens to the spectrum when we consider A+n^-γ G_n, G_n is a Ginibre matrix and γ>1/2? We will resent several theorems, specific examples, and fun experiments. Certain transfer matrices will play an important role. Based on joint work with Anirban Basak and Elliot Paquette.

Martin Zerner, Recurrence and transience of some contractive Markov chains with super-heavy tailed innovations in random environment

We characterize recurrence and transience of nonnegative multivariate autoregressive processes of order one with random contractive coefficient matrix, of subcritical multitype Galton-Watson branching processes in random environment with immigration, and of the related max-autoregressive processes and general random exchange processes. Our criterion is given in terms of the maximal Lyapunov exponent of the coefficient matrix and the cumulative distribution function of the innovation/immigration component. We apply the result to certain frog processes and excited random walks.

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