Fluctuations of linear statistics of half-heavy-tailed random 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 $N^{-1}\mathrm{Tr}\phi (A)$. 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 $N^{-\alpha /4}$.

24 pages. Fixed some typos in the previous version

References in corpus (3)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Spectrum of large random reversible Markov chains: Heavy-tailed weights on the complete graph