Interaction-correlated random matrices

Abstract

We introduce a family of random matrices where correlations between matrix elements are induced via interaction-derived Boltzmann factors. Varying these yields access to different ensembles. We find a universal scaling behavior of the finite-size statistics characterized by a heavy-tailed eigenvalue distribution whose extremes are governed by the Fréchet extreme value distribution for the case corresponding to a ferromagnetic Ising transition. The introduction of a finite density of nonlocal interactions restores standard random-matrix behavior. Suitably rescaled average extremes, playing a physical role as an order parameter, can thus discriminate aspects of the interaction structure; they also yield further nonuniversal information. In particular, the link between maximum eigenvalues and order parameters offers a potential route to resolving long-standing problems in statistical physics, such as deriving the exact magnetization scaling function in the two-dimensional Ising model at criticality.

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References in corpus (17)

• Machine learning and deep learning
• Level-Spacing Distributions and the Airy Kernel
• Noise Dressing of Financial Correlation Matrices
• Universal and non-universal properties of cross-correlations in financial time series
• Extreme value statistics of correlated random variables: a pedagogical review
• Spectra of Euclidean Random Matrices
• Topological and Dynamical Complexity of Random Neural Networks
• Statistics of Atmospheric Correlations
• Random Matrices with Slow Correlation Decay
• Random matrix analysis of complex networks
• Condensation and Extreme Value Statistics
• Stability of the Matrix Dyson Equation and Random Matrices with Correlations
• Random matrix ensembles from nonextensive entropy
• Breakdown of random-matrix universality in persistent Lotka--Volterra communities
• Edge Universality for Orthogonal Ensembles of Random Matrices
• Exact finite-size scaling for the random-matrix representation of bond percolation on square lattice
• Universal scaling and criticality of extremes in random matrix theory

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