Central limit theorems for random matrices: From resolvents to free probability

This thesis is structured into two parts. In the first part, we consider the random variable X := Tr(f1(W)A1 . . . fk(W)Ak) where W is an N × N Hermitian Wigner matrix, k ∈ N, and we choose (possibly N-dependent) regular functions f1, . . . , fk as well as bounded deterministic matrices A1, . . . , Ak. In this context, we prove a functional central limit theorem on macroscopic and mesoscopic scales, showing that the fluctuations of X around its expectation are Gaussian and that the limiting covariance structure is given by a deterministic recursion. We further give explicit error bounds in terms of the scaling of f1, . . . , fk and the number of traceless matrices among A1, . . . , Ak, thus extending the results of Cipolloni, Erdős and Schröder [40] to products of arbitrary length k ≥ 2. Analyzing the underlying combinatorics leads to a non-recursive formula for the variance of X as well as the covariance of X and Y := Tr(fk+1(W)Ak+1 . . . fk+ℓ(W)Ak+ℓ) of similar build. When restricted to polynomials, these formulas reproduce recent results of Male, Mingo, Peché, and Speicher [107], showing that the underlying combinatorics of noncrossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. As an application, we consider the fluctuation of Tr(eitW A1e −itW A2)/N around its thermal value Tr(A1) Tr(A2)/N2 when t is large and give an explicit formula for the variance. The second part of the thesis collects three smaller projects focusing on different random matrix models. In the first project, we show that a class of weakly perturbed Hamiltonians of the form Hλ = H0 + λW, where W is a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by Hλ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order λ −2 . As the main result, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent global law for the deformed Wigner matrix Hλ. The second project focuses on correlated random matrices, more precisely on a correlated N × N Hermitian random matrix with a polynomially decaying metric correlation structure. A trivial a priori bound shows that the operator norm of this model is stochastically dominated by √ N. However, by calculating the trace of the moments of the matrix and using the summable decay of the cumulants, the norm estimate can be improved to a bound of order one. In the third project, we consider a multiplicative perturbation of the form UA(t) where U is a unitary random matrix and A = diag(t, 1, ..., 1). This so-called UA model was first introduced by Fyodorov [73] for its applications in scattering theory. We give a general description of the eigenvalue trajectories obtained by varying the parameter t and introduce a flow of deterministic domains that separates the outlier resulting from the rank-one perturbation from the typical eigenvalues for all sub-critical timescales. The results are obtained under generic assumptions on U that hold for various unitary random matrices, including the circular unitary ensemble (CUE) in the original formulation of the model.