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Real symmetric random matrices and path counting.

Giovanni M Cicuta1

  • 1Dipartimento di Fisica, Universita di Parma, Parco Area delle Scienze 7A, 43100 Parma, Italy. cicuta@fis.unipr.it

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 4, 2005
PubMed
Summary
This summary is machine-generated.

This study evaluates expectations of matrix traces for real symmetric matrices with random entries. The findings offer insights into spectral density and entry rescalings in the large matrix limit.

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Area of Science:

  • * Mathematical Physics
  • * Random Matrix Theory

Background:

  • * Real symmetric matrices with independent and identically distributed (i.i.d.) random entries are fundamental in various scientific fields.
  • * Understanding the spectral properties of such matrices is crucial for analyzing complex systems.

Purpose of the Study:

  • * To derive exact evaluations for the trace of powers of random symmetric matrices (TrS(p)).
  • * To explore the behavior of spectral density in the large matrix limit (n -> infinity).
  • * To provide a tool for analyzing rescalings of matrix entries in the large n limit.

Main Methods:

  • * Exact evaluation of the trace of matrix powers, TrS(p), for real symmetric matrices S of order n.
  • * Analysis of matrix entries drawn from an arbitrary probability distribution.
  • * Polynomial expressions in terms of matrix entry moments are derived.

Main Results:

  • * Established that the expectations are polynomials in the moments of the matrix entries.
  • * Demonstrated the utility of these expectations for understanding spectral density in the large n limit.
  • * Showcased the straightforward application of the method to analyze entry rescalings.

Conclusions:

  • * The derived polynomial expressions offer precise insights into the spectral properties of random symmetric matrices.
  • * The methodology facilitates the study of spectral density and entry rescalings in the asymptotic regime.
  • * This work provides a valuable analytical tool for random matrix theory applications.