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Precise asymptotics for the spectral radius of a large random matrix

Giorgio Cipolloni, László Erdős, Yuanyuan Xu

Abstract

We consider the spectral radius of a large random matrix $X$ with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of $X$ in [29]. To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of $X-z$ for different complex shift parameters $z$ using the Dyson Brownian Motion.

Precise asymptotics for the spectral radius of a large random matrix

Abstract

We consider the spectral radius of a large random matrix with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of in [29]. To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of for different complex shift parameters using the Dyson Brownian Motion.
Paper Structure (19 sections, 30 theorems, 323 equations)

This paper contains 19 sections, 30 theorems, 323 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Summary of the methods of the proof
  4. Hermitization and local laws
  5. Proof strategy
  6. Step 1. Ginibre ensemble: Small $\eta$ integral over $[0,n^{-1+\epsilon}]$
  7. Step 2. i.i.d. ensemble: Integral over the small $\eta$ regime $[0,n^{-1+\epsilon}]$
  8. Step 3. i.i.d. ensemble: Large $\eta$ integral over $[n^{-1+\epsilon},T]$
  9. Third order terms with $p+q+1=3$ and with distinct indices $a\neq \underline{B}$ in \ref{['third_fourth_result']}
  10. Third order terms with index coincidence $a=\underline{B}$ in \ref{['third_fourth_result']}
  11. Fourth order terms with $p+q+1=4$ in \ref{['third_fourth_result']}
  12. Fifth order terms with $p+q+1=5$ in \ref{['third_fourth_result']}
  13. Weakly correlated Dyson Brownian motions at the cusp
  14. Proof of Theorem \ref{['local_thmw']}
  15. Proof of Lemma \ref{['lemma_third_order']}
  16. ...and 4 more sections

Key Result

Theorem 2.2

Let $X$ be an $n\times n$ matrix satisfyingThe matrix entries of $X$ do not have to be identically distributed. Our proof still works with minor modifications if ${\boldsymbol{E} } x_{ij}={\boldsymbol{E} } x_{ij}^2=0$, ${\boldsymbol{E} } |x_{ij}|^2=1/n$ and ${\boldsymbol{E} } |\sqrt{n}x_{ij}|^p \le Then we have for anyOur proof also gives an effective control on the probability in spectral_radiu

Theorems & Definitions (55)

  • Theorem 2.2
  • Remark 2.3
  • Theorem 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Lemma 3.4
  • proof : Proof of Theorem \ref{['main']}
  • Lemma 4.1
  • Proposition 4.2: Proposition 2.7 maxRe
  • proof : Proof of Lemma \ref{['lemma_step1']}
  • ...and 45 more