Asymptotic forms for hard and soft edge general β conditional gap probabilities Original Research Article

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 β-ensembles. The conditioning is that there are n eigenvalues in the gap, with image, 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 image. 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 image. They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating β to image.