[Paper Review] On the condition number of the critically-scaled Laguerre Unitary Ensemble

Percy Deift, Govind Menon+1 more|arXiv (Cornell University)|Jul 2, 2015

Random Matrices and Applications22 references4 citations

TL;DR

This paper establishes that under a critical scaling where $ n = N + \lfloor \sqrt{4cN} \rfloor $, the condition number $ \kappa $ of the Laguerre Unitary Ensemble (LUE) — a random Wishart matrix $ A = XX^* $ — converges in distribution to the Tracy–Widom distribution ($ \beta=2 $) as $ N \to \infty $. The result is derived via asymptotic analysis of Laguerre polynomials using Riemann–Hilbert problems, showing that extreme eigenvalue fluctuations (smallest, largest, and condition number) all follow the same universal Tracy–Widom law under this scaling, which is motivated by the conjugate gradient algorithm's halting time universality.

ABSTRACT

We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices $A = XX^*$, where $X$ is an $N imes n$ matrix with iid standard complex normal entries. Under the scaling $n = N + \lfloor \sqrt{ 4 c N} floor$, $c > 0$ and $N ightarrow \infty$, we show that the rescaled fluctuations of the smallest eigenvalue, largest eigenvalue and condition number of the matrices $A$ are all given by the Tracy--Widom distribution ($β= 2$). This scaling is motivated by the study of the solution of the equation $Ax=b$ using the conjugate gradient algorithm, in the case that $A$ and $b$ are random: For such a scaling the fluctuations of the halting time for the algorithm are empirically seen to be universal.

Motivation & Objective

• To establish the limiting distribution of the condition number $ \kappa = \lambda_{\max}/\lambda_{\min} $ for the Laguerre Unitary Ensemble (LUE) under a critical scaling of $ n = N + \lfloor \sqrt{4cN} \rfloor $.
• To demonstrate that the fluctuations of the smallest eigenvalue, largest eigenvalue, and condition number all converge to the Tracy–Widom distribution ($ \beta=2 $) in the $ N \to \infty $ limit.
• To connect this scaling to the empirical universality of the conjugate gradient algorithm's halting time in solving random linear systems $ Ax = b $.
• To rigorously derive the asymptotics of Laguerre polynomials via Riemann–Hilbert analysis to support the extreme eigenvalue limit theorems.

Proposed method

• Use of Riemann–Hilbert problem techniques to analyze the asymptotic behavior of Laguerre polynomials, which are fundamental to the eigenvalue density of the LUE.
• Application of classical Riemann–Hilbert analysis to derive precise asymptotics for the orthogonal polynomials associated with the LUE weight function.
• Derivation of the global eigenvalue density of the LUE and its connection to the Laguerre weight and orthogonal polynomial systems.
• Asymptotic analysis of the extreme eigenvalues using the derived polynomial asymptotics, focusing on edge scaling near the soft and hard edges of the spectrum.
• Establishment of the joint limit distribution of $ \lambda_{\min} $, $ \lambda_{\max} $, and $ \kappa $ via the Tracy–Widom law through careful scaling of $ \alpha = \lfloor \sqrt{4cN} \rfloor $.
• Use of Fredholm determinant representations involving the Airy kernel to express the Tracy–Widom distribution $ F_2(s) = \det(I - \mathcal{K}_{\text{Ai}}|_{L^2((s,\infty))}) $.

Experimental results

Research questions

• RQ1 Does the condition number of the LUE converge to the Tracy–Widom distribution under the critical scaling $ n = N + \lfloor \sqrt{4cN} \rfloor $?
• RQ2 How do the extreme eigenvalue fluctuations (smallest and largest) behave asymptotically in this scaling regime?
• RQ3 What is the connection between this scaling and the universality of the conjugate gradient algorithm’s halting time in random linear systems?
• RQ4 Can the joint distribution of $ \lambda_{\min} $, $ \lambda_{\max} $, and $ \kappa $ be described by a single universal law in the limit $ N \to \infty $?
• RQ5 What role does the $ \alpha \sim \sqrt{N} $ scaling play in avoiding heavy-tailed behavior of the condition number?

Key findings

• The smallest eigenvalue $ \lambda_{\min} $, when properly rescaled, converges in distribution to the Tracy–Widom distribution: $ \lim_{N\to\infty} \mathbb{P}\left( \frac{c - \lambda_{\min}}{c \alpha^{-2/3} 2^{2/3}} \leq t \right) = F_2(t) $.
• The largest eigenvalue $ \lambda_{\max} $, under the same scaling, also converges to the Tracy–Widom law: $ \lim_{N\to\infty} \mathbb{P}\left( \frac{\lambda_{\max} - \nu}{\nu^{1/3} 2^{2/3}} \leq t \right) = F_2(t) $, where $ \nu = 4N + 2\alpha + 2 $.
• The condition number $ \kappa = \lambda_{\max}/\lambda_{\min} $ converges to the Tracy–Widom distribution under the critical scaling: $ \lim_{N\to\infty} \mathbb{P}\left( \frac{\kappa - 4N/c}{4c^{-4/3} N^{2/3}} \leq t \right) = F_2(t) $.
• The critical scaling $ \alpha = \lfloor \sqrt{4cN} \rfloor $ ensures that $ \lambda_{\min} $ is pushed away from zero, eliminating heavy-tailed behavior and enabling pure Tracy–Widom fluctuations for $ \kappa $.
• The asymptotic analysis of Laguerre polynomials via Riemann–Hilbert problems provides the necessary foundation for deriving the extreme eigenvalue limits.
• The results explain the empirical universality of the conjugate gradient algorithm’s halting time in random systems: the critical scaling leads to a non-degenerate, universal limit for the convergence time.