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

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.

arXiv comment17 pages, 8 figures, Re-Submission to SciPost

References in corpus (9)

• Random-Matrix Theory of Quantum Transport
• A Random Matrix Approach to Cross-Correlations in Financial Data
• Extreme value statistics of correlated random variables: a pedagogical review
• Top eigenvalue of a random matrix: large deviations and third order phase transition
• Financial Applications of Random Matrix Theory: Old Laces and New Pieces
• Near-extreme eigenvalues and the first gap of Hermitian random matrices
• Finite-size critical scaling in Ising spin glasses in the mean-field regime
• Finite size effects and loss of self-averageness in the relaxational dynamics of the spherical Sherrington-Kirkpatrick model
• Annealed averages in spin and matrix models