ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR Β-Ensembles

Abstract

In a previous work [J. Math. Phys. 35 (1994) [2539][2540][2541][2542][2543][2544][2545][2546][2547][2548][2549][2550][2551], generalized hypergeometric functions have been used to a give a rigorous derivation of the large s asymptotic form of the general β > 0 gap probability E hard β (0; (0, s); βa/2), provided both βa/2 ∈ Z ≥0 and 2/β ∈ Z + . It is shown how the details of this method can be extended to remove the requirement that 2/β ∈ Z + . Furthermore, a large deviation formula for the gap probability E β (n; (0, x); ME β,N (λ aβ/2 e −2βNλ )) is deduced by writing it in terms of the characteristic function of a certain linear statistic. By scaling x = s/(4N ) 2 and taking N → ∞, this is shown to reproduce a recent conjectured formula for E hard β (n; (0, s); βa/2), βa/2 ∈ Z ≥0 , and moreover to give a prediction without the latter restriction. This extended formula, which for the constant term involves the Barnes double gamma function, is shown to satisfy an asymptotic functional equation relating the gap probability with parameters (β, n, a), to a gap probability with parameters (4/β, n , a ), where n = β(n + 1)/2 − 1, a = β(a − 2)/2 + 2.

Painlevé Representation of Tracy–Widom $${_\beta}$$ β Distribution for $${\beta}$$ β = 6

“…and, after taking into account that (ln F 0 ) ′ = Φ/3 and rescaling back t → 3 2/3 t, see the main theorem in the first section, this matches the known results [11], see also formula (2.16) in [24],…”

Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

“…A comprehensive account of the state of the art in this field, with formulation of the precise conjectures concerning the Dyson constants for general β-ensembles, can be found in the recent survey of Forrester and Sorrell [32].…”

Finite‐size corrections at the hard edge for the Laguerre β ensemble

“…is the limiting PDF of the smallest hard-edge scaled eigenvalue and # ( ) is the normalized histogram of the smallest eigenvalue from 10 10 numerically generated complex Wishart matrices scaled by the factor (4 + 2 ) with = 20, = 1. The red solid line is the predicted leading correction term of (1∕ 2 ) implied by either (23) or (37) substituted in (38).…”

“…It is shown in Ref. 37 that the variant of (47) given in Refs. 36 and 22 is then valid for general 2∕ > 0.…”