Convergence of Local Statistics of Dyson Brownian Motion

Abstract

Abstract:We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times t " op1q with deterministic initial data V . Our main result states that if the density of states of V is bounded both above and away from 0 down to scales ℓ ! t in a small interval of size G " t around an energy E 0 , then the local statistics coincide with the GOE/GUE near the energy E 0 after time t. Our methods are partly based on the idea of coupling two Dyson Brownian motions from [6], the parabolic regularity result of [15], and the eigenvalue rigidity results of [21].

Random matrices: tail bounds for gaps between eigenvalues

“…An analogue of (2) for real smooth ensembles, with the exponent k 2 replaced by k(k + 1)/2, was established in [4,Appendix B]. See also [15,18,19] for similar results. Finally, in [38], the last two authors established the bound…”

“…Additionally, arguing similarly to the proof of Theorem 6.3, if we let G i be the event that Conditioning on H i , it follows from (19), together with the small probability bound from Theorem 6.1 (applied to 2 d vectors), that…”

Local Semicircle Law for Random Regular Graphs

“…Here, the lower bound on the degree is of purely technical nature, and we believe that the results of [3] can be established with the same method under the weaker assumption d (log N ) O(1) . The local law proved in this paper, in addition to the results of [27,34], is an essential input for the proof in [3].…”

Local Law of Addition of Random Matrices on Optimal Scale

“…First, such bounds are used as a priori estimates in the proofs of Wigner-DysonMehta type universality results on local spectral statistics; see e.g. [12,20,21,27] and references therein. Second, control on the diagonal resolvent matrix elements for some η = Im z implies that the eigenvectors are delocalized on scale η −1 ; the optimal scale for η yields complete delocalization of the eigenvectors.…”