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Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model

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 f...

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Autores principales: de Freitas Pimenta, Pedro H., Stariolo, Daniel A.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2023
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10297421/
https://www.ncbi.nlm.nih.gov/pubmed/37372301
http://dx.doi.org/10.3390/e25060957
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author de Freitas Pimenta, Pedro H.
Stariolo, Daniel A.
author_facet de Freitas Pimenta, Pedro H.
Stariolo, Daniel A.
author_sort de Freitas Pimenta, Pedro H.
collection PubMed
description 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 are characterized by the two largest eigenvalues of a spike random matrix which defines the model, and particularly by the statistics concerning the gap between them. We characterize the finite-size statistics of the two largest eigenvalues 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 strength of the non-disordered perturbation in a way that is governed by the finite-size statistics of the gap.
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spelling pubmed-102974212023-06-28 Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model de Freitas Pimenta, Pedro H. Stariolo, Daniel A. Entropy (Basel) Article 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 are characterized by the two largest eigenvalues of a spike random matrix which defines the model, and particularly by the statistics concerning the gap between them. We characterize the finite-size statistics of the two largest eigenvalues 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 strength of the non-disordered perturbation in a way that is governed by the finite-size statistics of the gap. MDPI 2023-06-20 /pmc/articles/PMC10297421/ /pubmed/37372301 http://dx.doi.org/10.3390/e25060957 Text en © 2023 by the authors. https://creativecommons.org/licenses/by/4.0/Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
de Freitas Pimenta, Pedro H.
Stariolo, Daniel A.
Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model
title Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model
title_full Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model
title_fullStr Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model
title_full_unstemmed Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model
title_short Finite-Size Relaxational Dynamics of a Spike Random Matrix Spherical Model
title_sort finite-size relaxational dynamics of a spike random matrix spherical model
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10297421/
https://www.ncbi.nlm.nih.gov/pubmed/37372301
http://dx.doi.org/10.3390/e25060957
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