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

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

Random Matrices and Applications6 citations

TL;DR

This paper investigates the $\beta=2$ Stieltjes--Wigert random matrix ensemble, deriving exact expressions for its moments using little $q$-Jacobi polynomials and analyzing global and local scaling limits. It establishes that in an $L\to\infty$ limit, the edge scaling limit of the correlation kernel reduces to the classical Airy kernel, connecting $q$-special functions 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 $- \log |\sinh (π(x-y)/L) |$. 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 exact expressions for the moments of the $\beta=2$ Stieltjes--Wigert random matrix ensemble.
• To compute the global eigenvalue density from the large-$N$ form of these moments.
• To analyze the edge scaling limit of the correlation kernel and its relation to the Airy kernel.
• To establish the connection between the $q$-Airy function and the classical Airy function in a specific $L\to\infty$ scaling limit.
• To unify interpretations of the ensemble across non-intersecting Brownian motion, quantum many-body systems, and Chern-Simons theory.

Proposed method

• Use of the Stieltjes--Wigert polynomials to construct the correlation kernel of the determinantal point process.
• Derivation of exact moment formulas via a particular little $q$-Jacobi polynomial expression.
• Asymptotic analysis of the $q$-Airy function $A_q(z)$ using $q = e^{-\epsilon}$ and $\epsilon \to 0^+$, with $\alpha = \epsilon^{2/3}x$.
• Application of known asymptotic expansions from [31] to relate $A_q(z)$ to the Airy function $\mathrm{Ai}(x)$ and its derivative.
• Use of functional equations and special function identities, including the $q$-difference equation for $A_q(z)$.
• Transformation of the eigenvalue PDF via $u_j = q^{-N} e^{2\pi x_j / L}$ to map the system to a multiplicative $q$-deformation of the classical GUE.

Experimental results

Research questions

• RQ1 What is the exact form of the moments of the $\beta=2$ Stieltjes--Wigert random matrix ensemble?
• RQ2 How does the global eigenvalue density emerge from the large-$N$ behavior of these moments?
• RQ3 What is the limiting form of the edge scaling correlation kernel in the $L\to\infty$ limit?
• RQ4 How does the $q$-Airy function $A_q(z)$ asymptotically approach the classical Airy function $\mathrm{Ai}(x)$?
• RQ5 What is the connection between the $\beta=2$ Stieltjes--Wigert ensemble and the classical Airy kernel in the edge scaling limit?

Key findings

• The moments of the $\beta=2$ Stieltjes--Wigert ensemble are exactly expressed in terms of a little $q$-Jacobi polynomial.
• The global eigenvalue density is computed from the large-$N$ asymptotics of these moments.
• In the $L\to\infty$ scaling limit, the edge scaling limit of the correlation kernel reduces to the classical Airy kernel.
• The $q$-Airy function $A_q(z)$ asymptotically approaches the classical Airy function $\mathrm{Ai}(x)$ as $\epsilon \to 0^+$, with corrections involving $\mathrm{Ai}'(x)$.
• The asymptotic behavior of the gap probability $E^{(2d)}_{\rm edge}(0,(s,\infty))$ is shown to scale as $e^{-s^3 L / 24\pi}$ for large $s$, consistent with the Airy law.
• The functional equation $q x u(q^2 x) - u(q x) + u(x) = 0$ is satisfied by the $q$-Airy function $A_q(z)$, confirming its role in $q$-special function theory.