Biorthogonal Polynomials Related to Quantum Transport Theory of Disordered Wires

We consider the Plancherel–Rotach type asymptotics of the biorthogonal polynomials associated with the biorthogonal ensemble having the joint probability density function where In the special case where 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 the 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 theory, we establish Ohm’s law ( 1.13 ) and universal conductance fluctuation ( 1.14 ) for the disordered wire model, thereby rigorously confirming predictions from experimental physics (Washburn and Webb: Adv Phys 35(4):375–422, 1986).