Largest eigenvalue statistics of sparse random adjacency matrices

Abstract

We investigate the statistics of the largest eigenvalue, ${\lambda }_{\max }$, 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 ${\lambda }_{\max }$ 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 ${\lambda }_{\max }$ 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.

18 pages, 8 figures