Efficient computational method using random matrices describing critical thermodynamics

Roberto da Silva, Eliseu Venites Filho (), Sandra D. Prado () and J. R. D. de Felício ()
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Roberto da Silva: Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil
Eliseu Venites Filho: Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil
Sandra D. Prado: Instituto de Física, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil
J. R. D. de Felício: Departamento de Física, Faculdade de Filosofia, Ciências e Letras de Riberão Preto, Universidade de São Paulo, Ribeirão Preto, São Paulo, Brazil

International Journal of Modern Physics C (IJMPC), 2025, vol. 36, issue 01, 1-24

Abstract: Our study emphasizes the efficacy of employing matrices resembling Wishart matrices, derived from magnetization time series data within specific dynamics, to elucidate phase transitions and critical phenomena in the Q-state Potts model. Through the application of appropriate statistical methods, we not only identify second-order transitions but also distinguish weaker first-order transitions by carefully analyzing the density of eigenvalues and their fluctuations. Additionally, we investigate the method’s sensitivity to stronger first-order transition points. Notably, we establish a robust correlation between the system’s actual thermodynamics and the spectral thermodynamics encapsulated within the eigenvalues. Our findings are further supported by correlation histograms of the time series data, revealing insightful patterns. Building upon our core results, we provide a didactic analysis that draws parallels between the spectral properties of criticality in a spin system and matrices intentionally imbued with correlations (a toy model). Within this framework, we observe a universal behavior characterized by the distribution of eigenvalues into two distinct groups, separated by a gap dependent on the level of correlation, influenced by temperature-induced changes in the spin system.

Keywords: Random matrices; phase transitions; Wishart matrices; universal law deviations (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0129183124501638

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