[Paper Review] Global and local scaling limits for the $\beta = 2$ Stieltjes--Wigert random matrix ensemble

Peter J. Forrester|arXiv (Cornell University)|Nov 23, 2020

Random Matrices and Applications54 references4 citations

TL;DR

This paper investigates the β = 2 Stieltjes–Wigert random matrix ensemble, deriving exact expressions for its moments via little q-Jacobi polynomials and computing the global density in the large N limit. It establishes a scaling limit as L → ∞, showing that the edge scaling limit of the correlation kernel reduces to the classical Airy kernel, thereby connecting the q-deformed Stieltjes–Wigert system to universal random matrix behavior.

ABSTRACT

The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well known analogy with the Boltzmann factor for a classical log-gas with pair potential $- \log | x - y|$, confined by a one-body harmonic potential. A generalisation is to replace the pair potential by $- \log |\sinh (\pi (x-y)/L) |$. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time $t=0$, and subsequently in the study of quantum many body systems of the Calogero-Sutherland type, and also in Chern-Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes--Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little $q$-Jacobi polynomial. From their large $N$ form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ($q$-Airy) function. We show how in a particular $L o \infty$ scaling limit, this reduces to the Airy kernel.

Motivation & Objective

• To derive an exact expression for the moments of the β = 2 Stieltjes–Wigert random matrix ensemble.
• To compute the global density of states from the large N asymptotics of the moments.
• To analyze the edge scaling limit of the correlation kernel in a specific L → ∞ limit.
• To demonstrate the convergence of the q-Airy function to the classical Airy function in this limit.
• To establish the universality of the Airy kernel in the edge scaling limit of the Stieltjes–Wigert ensemble.

Proposed method

• Derives the moment-generating function using a particular little q-Jacobi polynomial expression.
• Applies large N asymptotic analysis to extract the global density from the moments.
• Utilizes the known edge scaling limit of the correlation kernel in terms of the Ramanujan (q-Airy) function.
• Employs asymptotic expansions of the q-Airy function Aq(z) as q → 1− (i.e., ϵ → 0+ with q = e−ϵ).
• Performs a scaling limit in the parameter L → ∞, relating the edge variables to the classical Airy kernel via functional equations and asymptotic expansions.
• Uses the functional equation and q-hypergeometric representation of Aq(z) to connect q-deformed behavior to the classical Airy function.

Experimental results

Research questions

• RQ1 What is the exact form of the moments of the β = 2 Stieltjes–Wigert random matrix ensemble?
• RQ2 How does the global density of states emerge from the large N limit of the moments?
• RQ3 What is the behavior of the edge scaling limit of the correlation kernel in the L → ∞ limit?
• RQ4 Does the q-Airy function Aq(z) converge to the classical Airy function under an appropriate scaling?
• RQ5 Can the universal Airy kernel be recovered as a limiting case of the Stieltjes–Wigert edge kernel?

Key findings

• The moments of the β = 2 Stieltjes–Wigert ensemble are exactly expressed in terms of a little q-Jacobi polynomial.
• The global density of the eigenvalue distribution is derived from the large N asymptotics of these moments.
• In the L → ∞ scaling limit, the edge scaling limit of the correlation kernel reduces to the classical Airy kernel.
• The asymptotic behavior of the q-Airy function Aq(z) as ϵ → 0+ (q = e−ϵ) is shown to reproduce the Airy function up to a prefactor and correction terms.
• The convergence of the q-Airy function to the classical Airy function is established via a precise asymptotic formula involving Ai and Ai′ functions with scaled arguments.
• The edge probability distribution in the 2D Coulomb gas model on a cylinder asymptotically decays as e−s³L/24π, consistent with the Airy law in the scaling limit.