Functional form for the leading correction to the distribution of the largest eigenvalue in the GUE and LUE

Abstract

The neighbourhood of the largest eigenvalue λmax in the Gaussian unitary ensemble (GUE) and Laguerre unitary ensemble (LUE) is referred to as the soft edge. It is known that there exists a particular centring and scaling such that the distribution of λmax tends to a universal form, with an error term bounded by 1/N2/3. We take up the problem of computing the exact functional form of the leading error term in a large N asymptotic expansion for both the GUE and LUE - two versions of the LUE are considered, one with the parameter a fixed and the other with a proportional to N. Both settings in the LUE case allow for an interpretation in terms of the distribution of a particular weighted path le..