TY - JOUR
AB - The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.
AU - Bao, Zhigang
AU - Erdös, László
AU - Schnelli, Kevin
ID - 1207
IS - 3
JF - Communications in Mathematical Physics
SN - 00103616
TI - Local law of addition of random matrices on optimal scale
VL - 349
ER -
TY - JOUR
AB - We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deter- ministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015).
AU - Ferrari, Patrik
AU - Nejjar, Peter
ID - 447
JF - Revista Latino-Americana de Probabilidade e Estatística
TI - Fluctuations of the competition interface in presence of shocks
VL - 9
ER -
TY - JOUR
AB - We consider random matrices of the form H=W+λV, λ∈ℝ+, where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d.\ entries that are independent of W. We assume subexponential decay for the matrix entries of W and we choose λ∼1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+∈ℝ+ such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ>λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N→∞ if λ>λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ>λ+, while they are completely delocalized for λ<λ+. Similar results hold for the lowest eigenvalues.
AU - Lee, Jioon
AU - Schnelli, Kevin
ID - 1881
IS - 1-2
JF - Probability Theory and Related Fields
TI - Extremal eigenvalues and eigenvectors of deformed Wigner matrices
VL - 164
ER -
TY - JOUR
AB - We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.
AU - Bourgade, Paul
AU - Erdös, László
AU - Yau, Horngtzer
AU - Yin, Jun
ID - 1280
IS - 10
JF - Communications on Pure and Applied Mathematics
TI - Fixed energy universality for generalized wigner matrices
VL - 69
ER -
TY - JOUR
AB - We prove that the system of subordination equations, defining the free additive convolution of two probability measures, is stable away from the edges of the support and blow-up singularities by showing that the recent smoothness condition of Kargin is always satisfied. As an application, we consider the local spectral statistics of the random matrix ensemble A+UBU⁎A+UBU⁎, where U is a Haar distributed random unitary or orthogonal matrix, and A and B are deterministic matrices. In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎A+UBU⁎ concentrates around the free additive convolution of the spectral distributions of A and B on scales down to N−2/3N−2/3.
AU - Bao, Zhigang
AU - Erdös, László
AU - Schnelli, Kevin
ID - 1434
IS - 3
JF - Journal of Functional Analysis
TI - Local stability of the free additive convolution
VL - 271
ER -