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Large gaps of CUE and GUE

Renjie Feng, Dongyi Wei

Abstract

In this article, we study the largest gaps of the classical random matrices of CUE and GUE, and show that after rescaling, the limiting densities are given by the Gumbel distributions.

Large gaps of CUE and GUE

Abstract

In this article, we study the largest gaps of the classical random matrices of CUE and GUE, and show that after rescaling, the limiting densities are given by the Gumbel distributions.

Paper Structure

This paper contains 16 sections, 16 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. A criterion for the Poisson convergence
  3. The CUE case
  4. A rescaling limit
  5. The strategy to prove Theorem \ref{['thm1']}
  6. The proof of Theorem \ref{['thm1']}
  7. An equivalent condition
  8. Upper bound
  9. Lower bound
  10. The GUE case
  11. Another rescaling limit
  12. One integral lemma
  13. The strategy to prove Theorem \ref{['thm2']}
  14. The proof of Theorem \ref{['thm2']}
  15. Upper bound
  16. ...and 1 more sections

Key Result

Theorem 1

Let's denote $m_k$ as the $k$-th largest gap of CUE and then the number of the rescaling point process $\{\tau_k\}_{k=1}^n$ falling in $[x,+\infty)$ tends to a Poisson random variable with mean $e^{c_1-x}$ for any fixed $x\in\mathbb R$. Here, $c_1=c_0+\ln(\pi/2)$ where $c_0=\frac{1}{12}\ln 2+3\zeta'(-1)$ and $\zeta(x)$ is the Riemann zeta function. This In particular, the limiting density for th

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • ...and 17 more