Biorthogonal polynomials related to quantum transport theory of disordered wires

Abstract

We consider the Plancherel-Rotach type asymptotics of the biorthogonal polynomials associated to the biorthogonal ensemble with the joint probability density function \begin{equation*} \frac{1}{C} \prod_{1 \leq i < j \leq n} (\lambda_j -\lambda_i)(f(\lambda_j) - f(\lambda_i)) \prod^n_{j = 1} W^{(n)}_{\alpha}(\lambda_j) d\lambda_j, \end{equation*} where \begin{align*} f(x) = {}& \sinh^2(\sqrt{x}), & W^{(n)}_{\alpha}(x) = {}& x^{\alpha} h(x) e^{-nV(x)}. \end{align*} In the special case that the potential function $V$ is linear, this biorthogonal ensemble arises in the quantum transport theory of disordered wires. We analyze the asymptotic problem via $2$-component vector-valued Riemann-Hilbert problems, and solve it under the one-cut regular with a hard edge condition. We use the asymptotics of biorthogonal polynomials to establish sine universality for the correlation kernel in the bulk, and provide a central limit theorem with a specific variance for holomorphic linear statistics. As an application of our theories, we establish the Ohm's law (1.12) and universal conductance fluctuation (1.13) for the disordered wire model, thereby rigorously confirming predictions from experimental physics [Washburn-Webb86].