Finite-size relaxational dynamics of a spike random matrix spherical model

Abstract

We present a thorough numerical analysis of the relaxational dynamics of the Sherrington-Kirkpatrick spherical model with an additive non-disordered perturbation for large but finite sizes $N$. In the thermodynamic limit and at low temperatures, the perturbation is responsible for a phase transition from a spin glass to a ferromagnetic phase. We show that finite size effects induce the appearance of a distinctive slow regime in the relaxation dynamics, the extension of which depends on the size of the system and also on the strength of the non-disordered perturbation. The long time dynamics is characterized by the two largest eigenvalues of a spike random matrix which defines the model, and particularly by the statistics of the gap between them. We characterize the finite size statistics of the two largest eignevalues of the spike random matrices in the different regimes, sub-critical, critical and super-critical, confirming some known results and anticipating others, even in the less studied critical regime. We also numerically characterize the finite size statistics of the gap, which we hope may encourage analytical work which is lacking. Finally, we compute the finite size scaling of the long time relaxation of the energy, showing the existence of power laws with exponents that depend on the strenght of the non-disordered perturbation, in a way which is governed by the finite size statistics of the gap.

arXiv comment14 pages, 10 figures

References in corpus (15)

• Level-Spacing Distributions and the Airy Kernel
• On Orthogonal and Symplectic Matrix Ensembles
• The largest eigenvalue of rank one deformation of large Wigner matrices
• Extreme value statistics of correlated random variables: a pedagogical review
• Full dynamical solution for a spherical spin-glass model
• Limits of spiked random matrices I
• Topology trivialization and large deviations for the minimum in the simplest random optimization
• Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model
• Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with ferromagnetic interaction
• Fundamental limits of detection in the spiked Wigner model
• Large time zero temperature dynamics of the spherical p=2-spin glass model of finite size
• Right tail expansion of Tracy-Widom beta laws
• The density of eigenvalues seen from the soft edge of random matrices in the Gaussian beta-ensembles
• Finite size effects and loss of self-averageness in the relaxational dynamics of the spherical Sherrington-Kirkpatrick model
• Free energy fluctuations of the $2$-spin spherical SK model at critical temperature