Biorthogonal polynomials related to quantum transport theory of disordered wires

TL;DR

The paper develops a vector RH framework to analyze biorthogonal polynomials tied to a biorthogonal ensemble with a nonlinear transform $f$ and linear-like potential weights, proving Plancherel-Rotach type asymptotics. It shows sine universality in the bulk, Airy and Bessel edge scalings at the soft and hard edges, respectively, and constructs explicit equilibrium measures under one-cut regularity with a hard edge. The authors also connect these asymptotics to statistical properties of linear statistics, establishing a CLT with a variance given by a double contour integral, and apply the results to quantum transport in disordered wires to rigorously confirm Ohm's law and universal conductance fluctuations. The methodology blends a two-component Riemann-Hilbert problem with global and local parametrices, providing a rigorous bridge between random matrix theory and mesoscopic physics. The findings solidify theoretical predictions for conductance phenomena in disordered wires and highlight robust bulk universality in biorthogonal ensembles.

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} (λ_j -λ_i)(f(λ_j) - f(λ_i)) \prod^n_{j = 1} W^{(n)}_α(λ_j) dλ_j, \end{equation*} where \begin{align*} f(x) = {}& \sinh^2(\sqrt{x}), & W^{(n)}_α(x) = {}& x^α 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].

Figures (9)

• Figure 1: Shape of $\mathbb{P}$ (the region to the right of the parabola).
• Figure 2: The schematic illustration of $J_c$ and $\mathbf{J}_{c}$ on $\mathbb{C}_+$. (The definitions of $J_c$ and $\mathbf{J}_{c}$ are extended to $\mathbb{C}_-$ naturally by complex conjugation.) If $c$ is replaced by a general $x \in (0, \infty)$, then $J_x(s_2(x)) = 2\sqrt{\mathfrak{b}(x)}$ and $\mathbf{J}_{x}(s_2(x)) = \mathfrak{b}(x)$ will change, while $J_x(s_1(x)) = \mathbf{J}_{x}(s_1(x)) = 0$, $J_x(0) = \pi i$, and $\mathbf{J}_{x}(0) = -\pi^2/4$ remain unchanged.
• Figure 3: $\mathbf{J}_{c}$ maps $\mathbb{C} \setminus \overline{D}$ to $\mathbb{C} \setminus [0, b]$ and maps $D \setminus [0, 1]$ to $\mathbb{P} \setminus [0, b]$.
• Figure 4: The lens $\Sigma$.
• Figure 5: Contour $\Sigma^R$.

Theorems & Definitions (48)

• Proposition 1.1
• Proposition 1.2
• Proposition 1.3
• Remark 1
• Remark 2
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8