In this paper we study multi-matrix models whose potentials are perturbations of the quadratic potential associated with independent GUE random matrices. More precisely, we compute the free energy and the expectation of the trace of polynomials evaluated in those matrices. We prove an asymptotic expansion in the inverse of the matrix dimension to any order. Out of this result we deduce new formulas for map enumerations and the microstates free entropy. Our approach is based on the interpolation method between random matrices and free operators developed in Collins et al. (Camb J Math 10: 195–260, 2022) and Parraud (Commun Math Phys 399: 1–46, 2022).

We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to N , the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in [9]. More precisely given P a self-adjoint non -commutative polynomial and Y N a d -tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator P ( Y N ) yields asymptotic freeness for large times.

We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix X∗X converge to its Tracy–Widom limit at a rate nearly N-1/3, where X is an M × N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate N-2/9 obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.

We show that the fluctuations of the largest eigenvalue of any generalized Wigner matrix H converge to the Tracy–Widom laws at a rate nearly O(N−1/3), as the matrix dimension N tends to infinity. We allow the variances of the entries of H to have distinct values but of comparable sizes such that (formula presented). Our result improves the previous rate O(N−2/9) by Bourgade [8] and the proof relies on the first long-time Green function comparison theorem near the edges without the second moment matching restriction.

We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws at a rate O(N-1/3+ω) , as N tends to infinity. For Wigner matrices this improves the previous rate O(N-2/9+ω) obtained by Bourgade (J Eur Math Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515, 2012) to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles. 

We consider N by N deformed Wigner random matrices of the form X-N = H-N + A(N), where H-N is a real symmetric or complex Hermitian Wigner matrix and A(N) is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of X-N for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on studying the characteristic function of the linear statistics (Landon and Sosoe (2018)) by using the cumulant expansion method, along with local laws for the Green function of X-N (Ann. Probab. 48 (2020) 963-1001; Probab. Theory Related Fields 169 (2017) 257-352; J. Math. Phys. 54 (2013) 103504) and analytic subordination properties of the free additive convolution (Dallaporta and Fevrier (2019); Random Matrices Theory Appl. 9 (2020) 2050011). We also prove the analogous results for high-dimensional sample covariance matrices.

We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices.

We consider random matrices of the form H-N = A(N) + UNBNUN*, where A(N) and B-N are two N by N deterministic Hermitian matrices and U-N is a Haar distributed random unitary matrix. We establish a universal central limit theorem for the linear eigenvalue statistics of H-N on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics and consists of two main steps: (1) generating Ward identities using the left-translation invariance of the Haar measure, along with a local law for the resolvent of H-N and analytic subordination properties of the free additive convolution, allows us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations.

We consider the free additive convolution of two probability measures mu and nu on the real line and show that mu boxed plus nu is supported on a single interval if mu and nu each has single interval support. Moreover, the density of mu boxed plus nu is proven to vanish as a square root near the edges of its support if both mu and nu have power law behavior with exponents between -1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5].

We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4,5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.