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

Abstract

We investigate the asymptotic behavior of the Selberg-like integral $$\frac{1}{N!}{\int }_{[0,1{]}^N}{x}_1^p\prod _{i<j}(x_i-x_j{)}^2\prod _i{x}_i^{a-1}(1-x_i{)}^{b-1}d{x}_i$$, 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.

References in corpus (8)

• Distributions of Conductance and Shot Noise and Associated Phase Transitions
• Product of random projections, Jacobi ensembles and universality problems arising from free probability
• Probability distributions of Linear Statistics in Chaotic Cavities and associated phase transitions
• Systematic approach to statistics of conductance and shot-noise in chaotic cavities
• Nonlinear statistics of quantum transport in chaotic cavities
• Statistics of quantum transport in chaotic cavities with broken time-reversal symmetry
• Full counting statistics of chaotic cavities with many open channels
• Nonlinear Random Matrix Statistics, symmetric functions and hyperdeterminants

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