Asymptotic forms for hard and soft edge general beta conditional gap probabilities

作者：Forrester Peter J; Witte Nicholas S^*

来源：Nuclear Physics B, 2012, 859(3): 321-340.

DOI：10.1016/j.nuclphysb.2012.02.008

An infinite log-gas formalism, due to Dyson, and independently Fogler and Shklovskii, is applied to the computation of conditioned gap probabilities at the hard and soft edges of random matrix beta-ensembles. The conditioning is that there are n eigenvalues in the gap, with n << vertical bar t vertical bar, t denoting the end point of the gap. It is found that the entropy term in the formalism must be replaced by a term involving the potential drop to obtain results consist with known asymptotic expansions in the case n = 0. With this modification made for general n, the derived expansions - which are for the logarithm of the gap probabilities - are conjectured to be correct up to and including terms O(log vertical bar t vertical bar). They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating beta to 4/beta.

• 出版日期2012-6-21