Asymptotics of Selberg-like integrals by lattice path counting

Abstract

We obtain explicit expressions for positive integer moments of the probability density of eigenvalues of the Jacobi and Laguerre random matrix ensembles, in the asymptotic regime of large dimension. These densities are closely related to the Selberg and Selberg-like multidimensional integrals. Our method of solution is combinatorial: it consists in the enumeration of certain classes of lattice paths associated to the solution of recurrence relations

Moments of the transmission eigenvalues, proper delay times, and random matrix theory. I

“…5), while (30) and (29) were recently rederived through combinatorial techniques. 45 Our exact results allow a simple derivation of these facts, while also consenting the investigation of β-dependent subleading corrections. We address these issues more thoroughly in the second part of this work, 41 where we show that the first two subleading terms in the asymptotic expansions of the previous theorems agree with those obtained semiclassically by Berkolaiko and Kuipers.…”

Large-N expansion for the time-delay matrix of ballistic chaotic cavities

“…The sum over diagrams is usually the hard part of the semiclassical approach; however, since the contribution of each diagram is given by the number of scattering channels N to some integer power, it is clear that the coefficients in the 1/N-expansion of τ (β) k are integers. We also mention that Novaes [29] computed the leading order of τ (β) k by considering the asymptotics of Selberg-like integrals; his method consists in enumerating certain classes of lattice paths. As expected, those paths were found to be in bijection with Schröder paths.…”

Large deviation eigenvalue density for the soft edge Laguerre and Jacobi β-ensembles

“…The derivation of the first non-oscillatory correction term for the Gaussian β-ensemble density given in [13, §14.3] implies that in the case of the Jacobi β-ensemble with weight as specified below (1.5), (2.6) can be extended to read To carry out this task, let the first term in (3.22) be denoted by Nρ J (1),∞ (x). We know from [31] (see also [27], [25]) that…”