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Time dependent fluctuations of linear eigenvalue statistics of some patterned matrices

Arup Bose, Shambhu Nath Maurya, Koushik Saha

Abstract

Consider the $n \times n$ reverse circulant $RC_n(t)$ and symmetric circulant $SC_n(t)$ matrices with independent Brownian motion entries. We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of these matrices as $n \tends \infty$, when the test functions of the statistics are polynomials. The proofs are mainly combinatorial, based on the trace formula, method of moments and some results on process convergence.

Time dependent fluctuations of linear eigenvalue statistics of some patterned matrices

Abstract

Consider the reverse circulant and symmetric circulant matrices with independent Brownian motion entries. We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of these matrices as , when the test functions of the statistics are polynomials. The proofs are mainly combinatorial, based on the trace formula, method of moments and some results on process convergence.

Paper Structure

This paper contains 7 sections, 15 theorems, 200 equations.

Table of Contents

  1. introduction and main results
  2. Proof of Theorem \ref{['thm:revcovar_rp']}
  3. Proof of Theorem \ref{['thm:revmulti_rp']}
  4. Proof of Theorem \ref{['thm:revprocess']}
  5. Proof of Theorem \ref{['thm:symcovar_sp']}
  6. Proof of Theorem \ref{['thm:symmulti_sp']}
  7. Proof of Theorem \ref{['thm:symprocess']}

Key Result

Theorem \oldthetheorem

For $0<t_1\leq t_2$ and $p,q\geq 1$, where

Theorems & Definitions (41)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • ...and 31 more