The neighbourhood of the largest eigenvalue λ max 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 λ max tends to a universal form, with an error term bounded by 1/N 2/3 . 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 length in a model involving exponential variables on a rectangular grid, as the grid size gets large. We give operator theoretic forms of the corrections, which are corollaries of knowledge of the first two terms in the large N expansion of the scaled kernel, and are readily computed using a method due to Bornemann. We also give expressions in terms of the solutions of particular systems of coupled differential equations, which provide an alternative method of computation. Both characterisations are well suited to a thinned generalisation of the original ensemble, whereby each eigenvalue is deleted independently with probability (1 − ξ). In the final section, we investigate using simulation the question of whether upon an appropriate centring and scaling a wider class of complex Hermitian random matrix ensembles have their leading correction to the distribution of λ max proportional to 1/N 2/3 . Here p U(N) (0; s) denotes the consecutive eigen-angle spacing distribution for Haar distributed unitary random matrices, with the angles rescaled to have mean spacing unity. On the RHS the quantity p 2 (0; s) -where the subscript "2" is the beta label from Dyson's three fold way [15] in the absence of time reversal symmetry -is the limiting distribution. Pioneering work in random matrix theory due to Mehta [37] and Gaudin [25] paved the way for Dyson [14] to obtain the Fredholm determinant formula2) where K s is the integral operator on 90, s) with kernel K(x, y) = sin π(x − y) π(x − y) . Two decades later the Kyoto school of Jimbo et al. [28] put (1.2) in the context of the Painlevé theory, obtaining the result det(I − ξK s ) = exp πs 0 σ (0) (t; ξ) t dt , (1.3)

The β ensembles are a class of eigenvalue probability densities which generalise the invariant ensembles of classical random matrix theory. In the case of the Gaussian and Laguerre weights, the corresponding eigenvalue densities are known in terms of certain β dimensional integrals. We study the large N asymptotics of the density with a soft edge scaling. In the Laguerre case, this is done with both the parameter a fixed, and with a proportional to N . It is found in all these cases that by appropriately centring the scaled variable, the leading correction term to the limiting density is O(N −2/3 ). A known differential-difference recurrence from the theory of Selberg integrals allows for a numerical demonstration of this effect. Date

A fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre β ensemble, characterized by the Dyson parameter β, and the Laguerre weight xae−βx/2, x>0 in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable x↦x/4N. Previous work has established the corresponding functional form of various statistical quantities—for example, the distribution of the smallest eigenvalue, provided that a∈double-struckZ≥0. We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling x↦x/4(N+a/β), the rate of convergence to the limiting distribution is O(1/N2), which is optimal. In the case β=2, general a>−1 the explicit functional form of the distribution of the smallest eigenvalue at this order can be computed, as it can for a=1 and general β>0. An iterative scheme is presented to numerically approximate the functional form for general a∈double-struckZ≥2.

Previous works have considered the leading correction term to the scaled limit of various correlation functions and distributions for classical random matrix ensembles and their β generalisations at the hard and soft edge. It has been found that the functional form of this correction is given by a derivative operation applied to the leading term. In the present work we compute the leading correction term of the correlation kernel at the spectrum singularity for the circular Jacobi ensemble with Dyson indices β = 1, 2 and 4, and also to the spectral density in the corresponding β-ensemble with β even. The former requires an analysis involving the Routh-Romanovski polynomials, while the latter is based on multidimensional integral formulas for generalised hypergeometric series based on Jack polynomials. In all cases this correction term is found to be related to the leading term by a derivative operation.