Random Schrödinger operators and related topics — abstracts

Jonathan Breuer, Spectral fluctuations for Schroedinger operators with a random decaying potential

Linear statistics have been a useful tool for studying fluctuations of eigenvalue counting measures in random matrix theory. In this talk we apply this tool to study fluctuations of this measure for discrete Schroedinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth rate of the variance of the corresponding linear statistic. In particular, each one of these subspaces defines a unique critical value for the decay-rate exponent, above which the random variable has a limit that is sensitive to the underlying distribution and below which the random variable has asymptotically Gaussian fluctuation. This is joint work with Yoel Grinshpon and Moshe White.

Margherita Disertori, Supersymmetric polar coordinates for random Schrödinger operators

Spectral properties of random Schrödinger operators can be derived from the average of products of Greens functions. For probability distributions with enough finite moments, the so-called supersymmetric approach offers a useful dual representation. We extend applicability of this method to less regular distributions, by using supersymmetric polar coordinates. This is joint work with M. Lager.

László Erdős, Central limit theorem for non-Hermitian random matrices

We prove that linear statistics for eigenvalues of non-Hermitian i.i.d. random matrices are Gaussian for both the complex and the real symmetry classes. The novel tools are (i) local laws for products of resolvents and (ii) analysis of Dyson Brownian motion (DBM) with correlations. The case of real matrices entail additional difficulties since the overlaps of the hermitized eigenvectors potentially jeopardise the well-posedness of the DBM. This is a joint work with Giorgio Cipolloni and Dominik Schroder.

Ilya Goldsheid, Exponential growth of products of non-stationary Markov-dependent matrices

I shall discuss sufficient conditions for exponential growth of the norm of a product of m × m matrices (with determinant 1) which are functions of the values of a non-homogeneous Markov chain. This generalizes the classical Furstenberg's theorem on products of independent identically distributed matrices as well as its extension by Virtser to the case of products of homogeneous Markov-dependent matrices.

Antti Knowles, Spectral analysis of critical Erdős-Rényi graphs

The Erdős-Rényi graph G(N,p) is the simplest model of a random graph, where each edge of the complete graph on N vertices is open with probability p, independently of the others. If p = pN is not too small then the degrees of the graph concentrate with high probability and the graph is homogeneous. On the other hand, for p of order (log N) / N and smaller, the degrees cease to concentrate and the graph is with high probability inhomogeneous, containing isolated vertices, leaves, hubs, etc. I present results on the eigenvalues and eigenvectors of the adjacency matrix of G(N,p) at and below this critical scale. I show a rigidity estimate for the locations of the eigenvalues and explain a transition from localized to delocalized eigenvectors at a specific location in the spectrum.

Cyril Labbé, The continuous Anderson Hamiltonian with a white noise

We will consider the Schrödinger operator obtained by perturbing the Laplacian with a white noise, on a finite box. In dimension 2 and 3, the operator needs to be renormalized by infinite constants and we will present a construction that relies on the theory of regularity structures. Then, in dimension 1, we will present results obtained in collaboration with Laure Dumaz (Paris-Dauphine) on the behavior of the spectrum when the size of the box is sent to infinity.

Bruno Nachtergaele, The transmission time and local integrals of motion for disordered spin chains

We investigate the relationship between zero-velocity Lieb-Robinson bounds and the existence of local integrals of motion (LIOMs) for disordered quantum spin chains. We also study the effect of dilute random perturbations on the dynamics of many-body localized spin chains. Using a notion of transmission time for propagation in quantum lattice systems we demonstrate slow propagation by proving a lower bound for the transmission time. This result can be interpreted as a robustness property of slow transport in one dimension. (Joint work with Jake Reschke)

Marcello Porta, A hierarchical supersymmetric model for weakly disordered 3d semimetals

As first pointed out by Wegner and Efetov in the 80s, noninteracting, disordered quantum lattice models can be mapped into interacting, nondisordered, supersymmetric quantum field theories. The mapping is exact, and it allows to represent, for instance, the averaged Green’s function of a random Schrödinger operator as the two point function of a supersymmetric lattice quantum field theory. The main obstacles in the rigorous study of such QFTs are: the large field problem, due to the unboundedness of the bosonic field; the infrared problem, arising every time we choose the Fermi level in the unperturbed spectrum; and finally, the oscillatory behavior of the purely imaginary covariance of the Gaussian SUSY field. In this talk I will focus on 3d semimetals, characterized by a pointlike Fermi surface. I will discuss the hierarchical approximation of the corresponding SUSY QFT; in particular, I will present a theorem about the algebraic decay of the two-point function of the model for weak interaction, corresponding to weak disorder. The proof of the theorem is based on the rigorous renormalization group, combined with stationary-phase methods and supersymmetric identities. Joint work with Giovanni Antinucci (University of Geneva) and Luca Fresta (University of Zurich).

Silke Rolles, Recent results on vertex-reinforced jump processes

Vertex-reinforced jump processes are stochastic processes in continuous time that prefer to jump to sites that have accumlated a large local time. Sabot and Tarr\`es showed interesting connections between vertex-reinforced jump processes and a supersymmetric hyperbolic nonlinear sigma model introduced by Zirnbauer in a completely different context. In the talk, I will present an extension of Zirnbauer's model and show how it arises naturally as a weak joint limit of a time-changed version of the vertex-reinforced jump process. It describes the asymptotics of rescaled crossing numbers, rescaled fluctuations of local times, asymptotic local times on a logarithmic scale, endpoints of paths, and last exit trees. Furthermore, I will present a construction of random interlacements for transient vertex-reinforced jump processes on a general graph. The talk is based on joint work with Franz Merkl and Pierre Tarrès.

Jeffrey Schenker, An ergodic theorem for homogeneously distributed quantum channels with applications to matrix product states

Quantum channels describe the most general form for the evolution of an open quantum system over a unit of time. Mathematically, a quantum channel is a completely positive and trace preserving linear map on the space of D × D matrices. We consider ergodic sequences of channels, obtained by sampling channel valued maps along the trajectories of an ergodic dynamical system. Repeated composition of these maps represents the result of repeated application of a given quantum channel subject to arbitrary correlated noise. Under a physically natural assumption, we obtain a general ergodic theorem showing that the composition of maps converges exponentially fast to a rank-one — entanglement breaking — channel. We apply this result to describe the thermodynamic limit of ergodic matrix product states and prove that correlations of observables in such states decay exponentially in the bulk. (Joint work with Ramis Movassagh)

Robert Seiringer, The polaron at strong coupling

We review old and new results on the Fröhlich polaron model. The discussion includes the validity of the (classical) Pekar approximation in the strong coupling limit, quantum corrections to this limit, as well as the divergence of the effective polaron mass.

Jacob Shapiro, Tight-Binding Reductions for Topological Insulators

We provide a scheme to derive topologically non-trivial tight-binding models out of continuum Schrodinger operators, and show that the topological indices between continuum and discrete levels match in the IQHE, Fu-Kane-Mele 2D, and chiral 1D cases; this provides another way to compute the Chern number or FKM Z2 index in the continuum. In the 1D case we discuss the question of defining a topological index before taking the tight-binding limit. Joint with M. I. Weinstein.

Avy Soffer, Random NLS Equations and Dispersive Theory of Liouvillian Dynamics

The Nonlinear Schroedinger equation with an attractive potential is considered, in the case where the potential is randomized by time dependent noise. It is shown that with probability 1 the solutions exist globally, and disperse, like free waves. This problem naturally leads to the dispersive analysis of Liouville dynamics of trace class operators, which is developed in this work, including Lp decay estimates. Joint work with Marius Beceanu.

Bálint Virág, The random planar geometry of the directed landscape

Random versions of Euclidean geometry in the plane are conjectured to have a universal scaling limit. The resulting random geometry is given by a random directed metric on the plane. I will explain the settings in which this limiting behavior has been proven, and show some properties of the limiting geometry, and how it is connected to models in the KPZ universality class. Joint work with Duncan Dauvergne, Mihai Nica and Janosch Ortmann.

Wei-Min Wang, Classical many body localized states

We discuss existence of classical many body localized states. These states are close to linear superpositions of eigenstates of the Anderson model for all time. In one dimension, this holds for all disorder. Eigenfunction labelling theorem and Minami estimates on eigenvalue spacing play an essential role.

Simone Warzel, The quantum random energy model

The quantum random energy model serves as a simple cornerstone, and a testing ground, for a number of fields. It is the simplest of all mean-field spin glass models in which quantum effects due to the presence of a transversal field are studied. In this talk I will review some of the conjectures related to this model and present a derivation of the key features of its thermal phase diagram.

Xiaolin Zeng, Reinforced loop soup isomorphism

We provide a Bayes formula which allows one to translate isomorphism theorems for simple random walks into their reinforced versions. First we discuss how one can translate the isomorphisms theorems for simple random walk into their vertex reinforced jump processes counterparts discovered by Bauerschmidt, Helmuth and Swan; then we recall simple random walk loop soup, and translate the loop soup isomorphism theorem to its reinforced counterpart. If time allows we will also discuss the duality between (reinforced) loop soup and Wilson's algorithm. Based on joint work with Chang, Liu and work in progress with Chang.

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