The random matrix hard edge: rare events and a transition

Abstract

We study properties of the point process that appears as the local limit at the random matrix hard edge. We show a transition from the hard edge to bulk behavior and give a central limit theorem and large deviation result for the number of points in a growing interval [0, λ] as λ → ∞. We study these results for the square root of the hard edge process. In this setting many of these behaviors mimic those of the Sine β process.

Exact large deviation function of spin current for the one dimensional XX spin chain with domain wall initial condition

“…Note added.-At the stage of the proofreading, we have realized that the large deviation function of the particle number in the hard edge is studied in a different fashion in[69]. The expression of large deviation in[69] seems different from ours but numerically looks almost the same.…”

Operator level limit of the circular Jacobi $β$-ensemble

“…A similar diffusion description for the square root of the hard edge process (spectrum of the operator G β,a given in ( 31)) was proved by Holcomb in [14], building on the results of [26]. Let M a,β (λ) be the counting function of the Bess β,a process and B a standard Brownian…”

“…A similar type of result is proved in [25] for the asymptotic gap probability of the hard edge process. Note also that for the square root of the hard edge process, various properties (for example a transition to Sine β process and a Central Limit Theorem) have been proved by Holcomb [14] by analyzing the coupled system of SDEs (40). With small modifications of the proofs therein, we get similar results for the HP β,δ process.…”

“…[34] provided a representation of the hard edge limit operator as a random Dirac operator. [14] provides various descriptions and properties of the limiting (hard edge) point process. Our main new contributions for the study of ROβE are the existence and description of the limit of the normalized characteristic polynomials (Corollary 17 and Theorem 23), and a new approach to prove the point process limit via operator convergence (Theorem 16).…”