The hole probability in log-gas and random matrix systems

Abstract

The probability of a particle free region (i.e. hole) in a hard-core lattice gas can be written as a finite sum over the distribution functions. In the special case in which the distribution functions have a determinant structure, of which the one-component log-gas at the couplings Γ = 2 and 4 on a one-dimensional lattice and the former system near a metal wall are examples, the sum can be written as a single Toeplitz determinant. Asymptotic formulas for the hole probability in the large hole size limit can then be obtained by using known theorems regarding the asymptotics of Toeplitz determinants. A conjecture, suggested by the exact analysis, is given for the leading behaviour of this prob..