Spectral Theory and pRObability in Mathematical physics (STROM)

Abstract: We discuss de/-localization in random and quasi-periodic Schroedinger operators and how this is related to the small-angle limit of twisted TMDs. In the case of disorder, our study is motivated by random displacements/strain, the quasi-periodicity arises when periodic Schroedinger operators are exposed to a perpendicular magnetic field.

Abstract: In this talk, we will introduce the Anderson hamiltonian, written as ``- Laplacian + random potential'', and discuss its spectral properties. It is well known that for a wide class of random potentials, localization occurs in one and two dimensions, while a major conjecture is to establish the existence of delocalization in three dimension.

We will focus here on the localization phenomenon in one dimension, in the specific case where the potential is given by the white noise. How does an electron localize in such a random environment? We will see that, as the energy increases, the eigenfunctions exhibit a simple shape, writing as the exponential of a Brownian motion with a drift, a form expected to be universal in one dimension for large energy or small potential limits.

Based on joint works with Cyril Labbé.

Abstract: Random matrices drawn from the Gaussian Ensembles represent systems with fully delocalized wave functions, referred to, in a broad sense, as "chaotic" in physics. We consider an open system, consisting of an internal chaotic system which is coupled to channels through which waves can enter and leave the internal system while interacting with it.

The setting just described is scattering in physics, indispensable to acquire experimental information on the quantum world. Universality is already present as we use a chaotic internal system modeled by a random matrix. This is the field of chaotic quantum scattering.

The unitary matrix mapping the in- and outgoing waves onto each other fully characterizes this open system, it is referred to as scattering matrix. In recent years we managed to solve an old problem and exactly calculated the distribution of the off-diagonal scattering matrix elements by putting forward a new variant of the supersymmetry method.

More than 60 years ago. Ericson conjectured that the real and imaginary parts of the scattering matrix are Gaussian distributed in the case that the coupling between in- and outgoing waves and internal delocalized states is strong while the mean level spacing of theses states becomes small. This is a common situation in quantum physics. Hence, another universality develops in universal chaotic quantum scattering. Very recently, we proved Ericson's conjecture.

Download Slides (PDF)

Abstract: The Rosenzweig-Porter (RP) model has recently gained a lot of attention as a paradigmatic (toy) model for studying localisation/delocalisation transitions.

In this talk, we report on a joint work with G. Cipolloni and L. Erdös, where we study the eigenvectors of a very general RP model, given by a Hamiltonian $H_\lambda = H_0 + \lambda W$. Here, $H_0$ is a completely arbitrary Hermitian deterministic matrix, $\lambda > 0$ an arbitrary coupling constant, and $W$ a random Wigner matrix. Our results include, in particular, a proof of a mobility edge (for certain $H_0$), a rigorous justification of a re-entrant localisation phenomenon (as recently put forward by Ghosh et al, PRB 2025), and a proof of a version of the Eigenstate Thermalisation Hypothesis (ETH).

To deduce these results on eigenvectors, we establish one- and two-resolvent local laws, which we prove by a dynamical method — the Zigzag strategy.

Abstract: On a compact manifold, the localization or delocalization of high-frequency Laplace eigenfunctions is strongly related to the properties of the geodesic flow. In this talk, we will be interested in delocalisation, through the study of $L^\infty$ norms of eigenfunctions on manifolds of negative curvature, whose geodesic flow is chaotic.

After recalling the existing results and conjectures, we will show how these results can be improved by adding small random perturbations to the Laplacian. We will also present deterministic improvements in constant negative curvature. These are joint works with Martin Vogel and Yann Chaubet.

Download Slides (PDF)

Abstract: I will discuss some results about the first nontrivial eigenvalue of the Laplace--Beltrami operator for a model of random coverings of a fixed surface of variable negative curvature. I will present some open questions about delocalization of eigenfunctions in this setting. Joint work with Will Hide and Frédéric Naud.

Abstract: In this talk, I shall review some of our recent results in three different disordered models with (i) flat band dispersions, (ii) many-body interactions and (iii) quasi-periodicity. In the flat band systems, I shall show how carefully selected choice of disorder can lead to seemingly delocalized behaviour even for very strong disorder. For quasi-periodicity, I show that the agreement with GOE statistics is excellent from a numerical point of view. For the disordered and interacting many-body systems, I shall review some recent numerical results on the issue of many-body localization in a high-symmetry situation.

Download Slides (PDF)

Abstract: We review recent results on the spectral and dynamical properties of Anderson models with long range hopping terms, including the so-called fractional Anderson model. In this talk we report on recent work with P. Hislop and R. Matos that shows that the local eigenvalue statistics for these models corresponds to Poisson statistics under certain conditions on the polynomial decay of the long-range hopping terms, which holds in a parameter region where one expects a conjecture on dynamical delocalization for such models to hold.

Abstract: Quantum mixing is a stronger version of quantum ergodicity introduced by Zelditch and Sunada, where the aim is to compare the asymptotics of observable averages of the form $\langle \psi_j, K \psi_k \rangle$ with uniform averages of the form $\delta_{j,k} \langle K \rangle_j$, where $\psi_j$ are the eigenfunctions of the underlying Laplacian. This gives information on both the asymptotic density $|\psi_j(x)|^2$ and the eigenfunction correlations $\psi_j(x)\psi_k(y)$.

We study this problem for sequences of Schreier graphs $G_N$ converging in the sense of Benjamini-Schramm to a Cayley graph which is spectrally delocalized. That is, the limiting Cayley graph $G$ should have some purely absolutely continuous spectrum in an interval $I$. We derive a general upper bound on the quantum variance, and use it to prove several results of quantum mixing. The more assumptions we make, the more observables $K$ we can control. With just AC spectrum in the limit, we can already prove mixing for all observables that are asymptotically uncorrelated. This covers random observables in particular. If we further assume a reasonable off-diagonal decay of the limiting resolvent, we can control all observables orthogonal to a subspace of dimension $o(N)$. If we further assume the $G_N$ converge strongly to $G$, we can control all observables.

The results go beyond regular graphs. I will give several examples of Cayley graphs we can handle (free products of finite rooted graphs, right-angled Coxeter groups...) and non-examples showing in some sense the necessity of our assumptions. One of the interests in the present results is that they cover limiting graphs which are neither trees nor $\mathbb{Z}^d$-periodic.

This is based on joint work with Charles Bordenave and Cyril Letrouit.

Abstract: Random Schroedinger operators are expected to have localized as well as delocalized eigenvectors. The former are associated with pure point spectrum, the latter with absolutely continuous spectrum. We review what is known about these two phases of disordered quantum matter from the Wegner-Efetov field theory formalism. Given the existence of singular continuous spectrum, which is associated with fractal-type eigenvectors, we inquire into the corresponding field-theory representation. Concrete results are presented for the Wegner N-orbital model in high space dimension.

Download Slides (PDF)