LOG-GASES, RANDOM MATRICES AND THE FISHER-HARTWIG CONJECTURE

Some features of the probability E(n, R) of a region R in certain log-potential systems containing precisely n particles are noted. First, it is shown that a quantity analogous to E(n, R) for a new solvable two-component log-gas can be expressed in terms of the Toeplitz determinant discretization of a Fredholm determinant which occurs in the calculation of E(n, R) for Hermitian random matrices. Second, the first two terms of the asymptotic large-R expansion of E(n, R) for complex random matrices, when R is a disk, are derived by using an electrostatic/thermodynamic argument based on an analogy with the two-dimensional one-component plasma. Finally, by using the Fisher-Hartwig 'conjecture' from the theory of Toeplitz determinants, the asymptotics of E(0, R) for a class of one-dimensional lattice systems is shown to obey a sum rule which has been conjectured to be applicable to all fluid systems with exclusively mobile species.