Mesoscopic central limit theorem for general $\beta$-ensembles

Abstract

We prove that the linear statistics of eigenvalues of β-log gasses satisfying the onecut and off-critical assumption with a potential V ∈ C 6 (R) satisfy a central limit theorem at all mesoscopic scales α ∈ (0; 1). We prove this for compactly supported test functions f ∈ C 5 (R) using loop equations at all orders along with rigidity estimates.

Microscopic description of log and Coulomb gases

“…• In dimension 1, this theorem was first proven in [Jo] for polynomial V and ξ analytic. It was later generalized in [Sh,BG1,BG2,BL,LLW], still with strong assumptions on ξ, and to the critical case in [BLS]. • If the extra conditions do not hold, then the CLT is not expected to hold.…”

“…The particular case of (1.2) for all β and general V , also called β-ensembles, has however been well-understood. In particular, thanks to the works of [Jo,Sh,BEY1,BEY2,BFG,BL], one has expansions of log Z N,β , Central Limit Theorems for linear statistics, and universality (after suitable rescaling) of the microscopic behavior and local statistics of the points, i.e. the fact that they are essentially independent of V .…”

Mesoscopic Linear Statistics of Wigner Matrices of Mixed Symmetry Class

“…Aside from the above articles, MLS had also been studied for invariant ensembles in [2,4,5,19,21], and for band matrices in [11,12].…”

Systems of Points with Coulomb Interactions

“…by numerical simulation, see Figure 3) suggest that the behavior of the system at the microscopic scale depends heavily on β, and one would like to describe this more precisely. In the particular case of (1.5) or (1.2) with β = 2, which both arise in Random Matrix Theory, many things can be computed explicitly, and expansions of log Z N,β as N → ∞, Central Limit Theorems for linear statistics, universality in V (after suitable rescaling) of the microscopic behavior and local statistics of the points, are known [Jo,Sh,BG1,BG2,BEY1,BEY2,BFG,BL]. Generalizing such results to higher dimensions and all β's is a significant challenge.…”

“…In dimension 1, this theorem was first proven in [Jo] for polynomial V and f analytic. It was later generalized in [Sh,BG1,BG2,BL,LLW,BLS]. In dimension 2, this result was proven for the determinantal case β = 2, first in [RV] (for V quadratic), [Ber3] assuming just f ∈ C 1 , and then [AHM] under analyticity assumptions.…”