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

Abstract

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 $n\ll |t|$, $t$ 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 consistent 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$(\log |t|)$. They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating $\beta$ to $4/\beta$.

Replaces v2 which contains typographical errors arising from a previous unpublished draft

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Cited by in corpus (4)

• Raney distributions and random matrix theory
• Large deviation eigenvalue density for the soft edge Laguerre and Jacobi $\beta$-ensembles
• Linear Differential Equations for the Resolvents of the Classical Matrix Ensembles
• Equilibrium measures for higher dimensional rotationally symmetric Riesz gases