ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR beta-ENSEMBLES

作者：Forrester Peter J^*

来源：Random Matrices-Theory and Applications, 2013, 2(2): 1350002.

DOI：10.1142/S2010326313500020

• Laguerre ensemble
• spacing distributions
• asymptotics
• Barnes double gamma function

In a previous work [J. Math. Phys. 35 (1994) 2539-2551], generalized hypergeometric functions have been used to a give a rigorous derivation of the large s asymptotic form of the general beta > 0 gap probability E-beta(hard)(0; ( 0, s); beta a/2), provided both beta a/ 2 is an element of Z >= 0 and 2/beta is an element of Z(+). It is shown how the details of this method can be extended to remove the requirement that 2/beta is an element of Z(+). Furthermore, a large deviation formula for the gap probability E-beta( n; (0, x); ME beta,N(lambda(a beta/2)e(-2 beta N lambda))) 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 ->infinity, this is shown to reproduce a recent conjectured formula for E-beta(hard)(n; (0, s); beta a/2), beta a/2 is an element of 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 ( beta, n, a), to a gap probability with parameters (4/beta, n', a'), where n' = beta(n + 1)/2 - 1, a' = beta(a - 2)/2 + 2.

• 出版日期2013-4