Largest eigenvalue statistics of sparse random adjacency matrices

TL;DR

This paper studies the distribution of the largest eigenvalue in large sparse adjacency matrices near the percolation threshold, revealing a universal Gumbel distribution and finite-size scaling behavior.

Contribution

It analytically and numerically demonstrates that the largest eigenvalue follows a Gumbel distribution near the percolation threshold in sparse matrices, uncovering a new universality.

Findings

Largest eigenvalue distribution approximates Gumbel near $p_c$

Finite-size corrections scale as $ o rac{1}{ ext{ln}^2 N}$

Universal eigenvalue behavior identified close to percolation threshold

Abstract

We investigate the statistics of the largest eigenvalue, $\lambda_{\rm max}$, in an ensemble of $N\times N$ large ($N\gg 1$) sparse adjacency matrices, $A_N$. The most attention is paid to the distribution and typical fluctuations of $\lambda_{\rm max}$ in the vicinity of the percolation threshold, $p_c=\frac{1}{N}$. The overwhelming majority of subgraphs representing $A_N$ near $p_c$ are exponentially distributed linear subchains, for which the statistics of the normalized largest eigenvalue can be analytically connected with the Gumbel distribution. For the ensemble of {\rm all} subgraphs near $p_c$ we suggest that under an appropriate modification of the normalization constant the Gumbel distribution provides a reasonably good approximation. Using numerical simulations we demonstrate that the proposed transformation of $\lambda_{\rm max}$ is indeed Gumbel-distributed and the leading finite-size corrections in the vicinity of $p_c$ scale with $N$ as $\sim \ln^{-2}N$. All together, our results reveal a previously unknown universality in eigenvalue statistics of sparse matrices close to the percolation threshold.