[Paper Review] Autocorrelations of the characteristic polynomial of a random matrix under microscopic scaling

Rowan Killip, E. Ryckman|arXiv (Cornell University)|Apr 9, 2010

Theoretical and Computational Physics23 references3 citations

TL;DR

This paper computes the microscopic-scale autocorrelation of the characteristic polynomial of a random matrix with eigenvalues distributed according to a β-ensemble on the unit circle. Using a novel method based on solving a system of linear ODEs, it derives an exact expression for the limit of the autocorrelation as the matrix size $ n \to \infty $, showing analytic dependence on $ \beta $, and proves that no phase transition occurs at the level of the characteristic polynomial, with the result expressed in terms of Bessel functions for general $ \beta \in (0,\infty) $.

ABSTRACT

We calculate the autocorrelation function for the characteristic polynomial of a random matrix in the microscopic scaling regime. While results fitting this description have be proved before, we will cover all values of inverse temperature $β\in (0,\infty)$. The method also differs from prior work, relying on matrix models introduced by Killip and Nenciu.

Motivation & Objective

• To compute the asymptotic autocorrelation of the characteristic polynomial of a random matrix in the microscopic scaling limit for general inverse temperature $ \beta \in (0,\infty) $.
• To extend prior results limited to $ \beta = 1,2,4 $ to the full continuous range of $ \beta $, using a new method not relying on determinantal structures.
• To investigate whether phase transitions occur in the characteristic polynomial statistics by analyzing the analyticity of the autocorrelation function in $ \beta $.
• To derive a general formula for higher-order autocorrelations involving multiple evaluation points near the unit circle, expressed as solutions to a system of linear ODEs.
• To demonstrate that the resulting autocorrelation is an analytic function of $ \beta $, $ w_j $, and $ y_k $, suggesting no phase transition at the level of the characteristic polynomial.

Proposed method

• Models eigenvalue statistics via the $ \beta $-ensemble on the unit circle with joint density proportional to $ |\Delta(e^{i\theta_1}, \dots, e^{i\theta_n})|^\beta $, where $ \beta $ is the inverse temperature.
• Introduces the characteristic polynomial $ Z_n(z) = \prod_{j=1}^n (1 - z^{-1} e^{i\theta_j}) $, normalized to remove the $ z^n $ factor for analytic number theory analogies.
• Applies a microscopic scaling by evaluating $ Z_n(e^{iw/n}) $ and $ Z_n(e^{iy/n}) $ as $ n \to \infty $, focusing on the typical inter-particle spacing.
• Derives the asymptotic autocorrelation as a solution to a system of linear ordinary differential equations (ODEs) with regular singular points, using a matrix-valued ODE framework.
• Solves the ODE system explicitly for the $ q=r=1 $ case using Bessel functions, leveraging known identities and Legendre's duplication formula for the Gamma function.
• Establishes analyticity of the autocorrelation in $ \beta $, $ w_j $, and $ y_k $, implying no phase transition at the level of the characteristic polynomial.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the autocorrelation $ \mathbb{E}^\beta_n\left[ Z_n(e^{ix/n}) \overline{Z_n(e^{-ix/n})} \right] $ in the microscopic limit as $ n \to \infty $, for general $ \beta \in (0,\infty) $?
• RQ2 Does the $ \beta $-ensemble characteristic polynomial exhibit a phase transition in its statistical properties, as suggested by non-analytic behavior in $ \beta $?
• RQ3 Can the higher-order autocorrelations $ \mathbb{E}^\beta_n\left[ \prod_{j=1}^q Z_n(e^{iw_j/n}) \prod_{k=1}^r \overline{Z_n(e^{iy_k/n})} \right] $ be computed for general $ \beta $, and what structure underlies their asymptotics?
• RQ4 How does the analytic structure of the autocorrelation function in $ \beta $ reflect the underlying universality or non-universality of the $ \beta $-ensemble?
• RQ5 What is the precise functional form of the microscopic autocorrelation for general $ \beta $, and how does it reduce to known results for $ \beta = 2 $ (Haar measure on $ U(n) $) or $ \beta = 1,4 $ (classical symmetric spaces)?

Key findings

• The microscopic autocorrelation $ \lim_{n\to\infty} n^{-2/\beta} \mathbb{E}^\beta_n\left[ Z_n(e^{iw/n}) \overline{Z_n(e^{iy/n})} \right] $ is given by $ \sqrt{\pi} \, e^{-i(w - \bar{y})/2} (w - \bar{y})^{1/2 - 2/\beta} J_{2/\beta - 1/2}\left( \frac{w - \bar{y}}{2} \right) $, where $ J_\nu $ is the Bessel function of the first kind.
• The result is analytic in all parameters $ \beta $, $ w $, and $ y $, indicating no phase transition in the characteristic polynomial statistics at the microscopic scale.
• For the special case $ w = x $, $ y = x $, the $ 2r $-th moment $ \lim_{n\to\infty} n^{-2r^2/\beta} \mathbb{E}^\beta_n\left[ |Z_n(e^{ix})|^{2r} \right] $ equals $ \prod_{p=1}^r \frac{\Gamma(2p/\beta)}{\Gamma(2(r+p)/\beta)} $, a finite product of Gamma functions.
• The method generalizes beyond $ \beta = 1,2,4 $, where determinantal/Pfaffian structures exist, to arbitrary $ \beta $, using ODEs instead of integrable systems.
• The solution to the ODE system for general $ R = q + r $ yields a function analytic in $ \beta $, supporting the absence of phase transitions in the characteristic polynomial's correlation structure.
• For higher-order correlations, the results are expressible as solutions to linear ODEs, though explicit closed forms involve exotic functions (e.g., Whittaker, hypergeometric), and their information content is not clearly superior to the ODE formulation.