Fluctuations of linear statistics of half-heavy-tailed random matrices

TL;DR

This paper investigates the fluctuation behavior of linear spectral statistics of Wigner matrices with entries having tail decay between heavy and light tails, revealing an intermediate fluctuation order depending on tail decay parameter.

Contribution

It fills the gap in understanding the fluctuation behavior of linear spectral statistics for matrices with tail decay exponent between 2 and 4.

Findings

Fluctuations scale as N^{-rac{eta}{4}} for 2<eta<4.

Identifies the intermediate fluctuation order bridging heavy and light tails.

Provides theoretical insight into spectral statistics of half-heavy-tailed matrices.

Abstract

We consider a Wigner matrix $A$ with entries tail decaying as $x^{-\alpha}$ with $2<\alpha<4$ for large $x$ and study fluctuations of linear statistics $N^{-1}\operatorname{Tr}\varphi(A)$. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. $\alpha < 2$) and light-tailed matrices (i.e. $\alpha > 4$). This paper fills in the gap of understanding for $2<\alpha<4$. We find that while linear spectral statistics for heavy-tailed matrices have fluctuations of order $N^{-1/2}$ and those for light-tailed matrices have fluctuations of order $N^{-1}$, the linear spectral statistics for half-heavy-tailed matrices exhibit an intermediate $\alpha$-dependent order of $N^{-\alpha/4}$.