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Cross-Modal Multivariate Pattern Analysis
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Identifying patterns using cross-correlation random matrices derived from deterministic and stochastic differential

Roberto da Silva1, Sandra D Prado1

  • 1Institute of Physics, Federal University of Rio Grande do Sul, Porto Alegre, Rio Grande do Sul 91501-970, Brazil.

Chaos (Woodbury, N.Y.)
|March 21, 2025
PubMed
Summary

Spectral properties of random matrices from maps can indicate critical or chaotic behavior in systems. This method avoids complex simulations for analyzing spin and chaotic systems.

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Area of Science:

  • Statistical Mechanics
  • Chaos Theory
  • Random Matrix Theory

Background:

  • Cross-correlation random matrices are indicators of phase transitions in spin systems.
  • Magnetization evolution in spin systems contains thermodynamic information reflected in matrix eigenvalues.
  • The Langevin equation's potential to capture chaotic behavior via maps is under investigation.

Purpose of the Study:

  • To propose using spectral properties of random matrices derived from differential equation maps.
  • To demonstrate this approach for identifying critical or chaotic system behavior.
  • To offer an alternative to traditional simulation methods.

Main Methods:

  • Constructing random matrices from maps derived from deterministic or stochastic differential equations.
  • Analyzing spectral properties of these matrices.
  • Utilizing iterated Hamiltonian equations for chaotic systems.
  • Employing Langevin maps from mean-field equations for spin systems.

Main Results:

  • The spectral properties of these random matrices effectively indicate critical behavior in spin systems.
  • The approach also identifies chaotic behavior in relevant systems.
  • This method bypasses the necessity for Monte Carlo (MC) simulations.

Conclusions:

  • Random matrix spectral analysis of derived maps is a viable tool for detecting critical and chaotic dynamics.
  • This technique offers a computationally efficient alternative to MC simulations in specific physical systems.