Exact finite-size scaling of the maximum likelihood spectra in the quenched and annealed Sherrington-Kirkpatrick spin glass

TL;DR

This paper develops an exact finite-size scaling analysis of the maximum likelihood spectra in the quenched and annealed SK spin glass models, revealing spectral crossover behavior and providing analytic predictions validated by simulations.

Contribution

It introduces a novel ODE-based method for analyzing spectral edge scaling in random matrices, specifically applied to the SK spin glass, and predicts finite-size effects in the spectral gap development.

Findings

Analytic predictions for spectral gap scaling in the annealed SK model.

Validation of scaling predictions through Monte Carlo simulations.

Identification of eigenvalue fluctuation effects near criticality.

Abstract

Fine resolution of the discrete eigenvalues at the spectral edge of an $N\times N$ random matrix is required in many applications. Starting from a finite-size scaling ansatz for the Stieltjes transform of the maximum likelihood spectrum, we demonstrate that the scaling function satisfies a first-order ODE of the Riccati type. Further transformation yields a linear second-order ODE for the characteristic function, whose nodes determine leading eigenvalues. Using this technique, we examine in detail the spectral crossover of the annealed Sherrington-Kirkpatrick (SK) spin glass model, where a gap develops below a critical temperature. Our analysis provides analytic predictions for the finite-size scaling of the spin condensation phenomenon in the annealed SK model, validated by Monte Carlo simulations. Deviation of scaling amplitudes from their predicted values is observed in the critical region due to eigenvalue fluctuations. More generally, rescaling the spectral axis, adjusted to the distance of neighboring eigenvalues, offers a powerful approach to handling singularities in the infinite size limit.