## Mid-Term Report Summary - RANMAT (Random matrices, universality and disordered quantum systems)

Random matrices have been extensively used in nuclear and solid state physics to describe local statistics of energy levels. More recently, random matrices appear in statistics, in wireless communication and in analysis of big data. These applications serve our main motivation in selecting our models that we analyze with full mathematical rigor.

The simplest random matrix model is the ensemble of Wigner matrices that are hermitian matrices with independent, identically distributed entries. Their density of states is given by the celebrated Wigner semicircle law and the local eigenvalue statistics is described by Dyson’s sine kernel. The first result was obtained by Wigner in the 50’s, the second result is the famous Wigner-Dyson-Mehta universality conjecture that was proven a few years ago by the PI and collaborators. Universality in this context means that local spectral statistics (e.g. distribution of gaps between neighboring eigenvalues) are independent of the details of the distribution of the Wigner matrix elements.

Wigner matrices describe the simplest mean field quantum system, where the random matrix elements correspond to quantum transition rates and every transition rate has the same probability distribution. One main achievement of the project is to extend the analysis to more general classes of random matrices. Two directions are especially important:

(i) remove the identical distribution requirement and

(ii) remove the independence requirement.

In a series of papers with Ajanki and Kruger we achieved both goals. First we gave a complete analysis of the density of states (that typically differs from the semicircle) and, second, we proved universality in the bulk spectrum.

We developed new tools for both steps. For the analysis of the density of states (and its local version, called the “local law”) we extensively analyzed the Matrix Dyson Equation (MDE). This equation was introduced at least 20 years ago as a computational tool to determine the density of states, but very little mathematical analysis was available. We proved bounds and regularity of the solution, as well as stability properties of the MDE. The main technical novelty is to perform this analysis for a spectral parameter very close to the real axis — this enabled us to use the result for local eigenvalue statistics.

For the universality, we generalized the approach of the Dyson Brownian motion to situations where the density of states is not the semicircle. In particular, the initial condition is far from global equilibrium, nevertheless, jointly with Schnelli, we managed to prove fast convergence to local equilibrium. This generalization was then combined with the three-step strategy developed earlier for the resolution of the WDM conjecture.

In an independent work (joint with Bourgade, Yau and Yin), we departed from the mean field character of Wigner’s model in the direction of the random banded matrices and we proved universality for a certain class of block band matrices, where the block sizes are still comparable with N, the dimension. In this work we overcame a basic restrictive assumption in the previous applications of the Dyson Brownian Motion, namely that every matrix element must have a nonzero variance.

For a related band matrix model, but with a narrower band, jointly with Bao we also proved local law on the optimal scale. This work used a rigorous supersymmetric analysis; a technique quite different from any previous approaches to local laws.

Another fundamental random matrix model is the free addition. If two hermitian matrices A and B describe two parts of the energy of a quantum system and we are interested in the total energy, we need to consider the sum A+B. However, A and B may not be given in the same basis, in fact in certain phenomenological models it is reasonable to assume that their relative basis is arbitrary. Modeling this base change with a Haar distributed unitary matrix U gives rise to the matrix H=A+UBU*. Using Voiculescu’s celebrated theory, the density of states of H can be determined as the free (additive) convolution of the spectral densities of A and B if the dimension is large. This theory is well developed on macroscopic scales (corresponding to a fixed spectral parameter that is independent of the size of the matrix), but very little was known about its local aspects, i.e. the analogues of the local laws. In a sequence of papers with Bao and Schnelli we developed the local theory of free addition and we proved local laws on the optimal scale. As a byproduct, we also obtained the optimal convergence 1/N rate in Voiculescu’s theorem.

The simplest random matrix model is the ensemble of Wigner matrices that are hermitian matrices with independent, identically distributed entries. Their density of states is given by the celebrated Wigner semicircle law and the local eigenvalue statistics is described by Dyson’s sine kernel. The first result was obtained by Wigner in the 50’s, the second result is the famous Wigner-Dyson-Mehta universality conjecture that was proven a few years ago by the PI and collaborators. Universality in this context means that local spectral statistics (e.g. distribution of gaps between neighboring eigenvalues) are independent of the details of the distribution of the Wigner matrix elements.

Wigner matrices describe the simplest mean field quantum system, where the random matrix elements correspond to quantum transition rates and every transition rate has the same probability distribution. One main achievement of the project is to extend the analysis to more general classes of random matrices. Two directions are especially important:

(i) remove the identical distribution requirement and

(ii) remove the independence requirement.

In a series of papers with Ajanki and Kruger we achieved both goals. First we gave a complete analysis of the density of states (that typically differs from the semicircle) and, second, we proved universality in the bulk spectrum.

We developed new tools for both steps. For the analysis of the density of states (and its local version, called the “local law”) we extensively analyzed the Matrix Dyson Equation (MDE). This equation was introduced at least 20 years ago as a computational tool to determine the density of states, but very little mathematical analysis was available. We proved bounds and regularity of the solution, as well as stability properties of the MDE. The main technical novelty is to perform this analysis for a spectral parameter very close to the real axis — this enabled us to use the result for local eigenvalue statistics.

For the universality, we generalized the approach of the Dyson Brownian motion to situations where the density of states is not the semicircle. In particular, the initial condition is far from global equilibrium, nevertheless, jointly with Schnelli, we managed to prove fast convergence to local equilibrium. This generalization was then combined with the three-step strategy developed earlier for the resolution of the WDM conjecture.

In an independent work (joint with Bourgade, Yau and Yin), we departed from the mean field character of Wigner’s model in the direction of the random banded matrices and we proved universality for a certain class of block band matrices, where the block sizes are still comparable with N, the dimension. In this work we overcame a basic restrictive assumption in the previous applications of the Dyson Brownian Motion, namely that every matrix element must have a nonzero variance.

For a related band matrix model, but with a narrower band, jointly with Bao we also proved local law on the optimal scale. This work used a rigorous supersymmetric analysis; a technique quite different from any previous approaches to local laws.

Another fundamental random matrix model is the free addition. If two hermitian matrices A and B describe two parts of the energy of a quantum system and we are interested in the total energy, we need to consider the sum A+B. However, A and B may not be given in the same basis, in fact in certain phenomenological models it is reasonable to assume that their relative basis is arbitrary. Modeling this base change with a Haar distributed unitary matrix U gives rise to the matrix H=A+UBU*. Using Voiculescu’s celebrated theory, the density of states of H can be determined as the free (additive) convolution of the spectral densities of A and B if the dimension is large. This theory is well developed on macroscopic scales (corresponding to a fixed spectral parameter that is independent of the size of the matrix), but very little was known about its local aspects, i.e. the analogues of the local laws. In a sequence of papers with Bao and Schnelli we developed the local theory of free addition and we proved local laws on the optimal scale. As a byproduct, we also obtained the optimal convergence 1/N rate in Voiculescu’s theorem.