Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula

TL;DR

This paper analyzes the asymptotic behavior of a Selberg-like integral relevant to random matrix theory, using Newton's interpolation formula to explicitly compute its limit as the number of variables grows large.

Contribution

It introduces a method employing Newton's interpolation formula to explicitly determine the asymptotic limit of Selberg-like integrals in the unitary case.

Findings

Asymptotic limit of the integral exists for large N

Explicit formula for the asymptotic limit derived

Method applicable to similar integrals in random matrix theory

Abstract

We investigate the asymptotic behavior of the Selberg-like integral $$ \frac1{N!}\int_{[0,1]^N}x_1^p\prod_{i<j}(x_i-x_j)^2\prod_ix_i^{a-1}(1-x_i)^{b-1}dx_i$$, as $N\to\infty$ for different scalings of the parameters $a$ and $b$ with $N$. Integrals of this type arise in the random matrix theory of electronic scattering in chaotic cavities supporting $N$ channels in the two attached leads. Making use of Newton's interpolation formula, we show that an asymptotic limit exists and we compute it explicitly.