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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 fluctuationsof $\lambda_{\rm max}$ 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 thestatistics of the normalized largest eigenvalue can be analytically connectedwith the Gumbel distribution. For the ensemble of {\rm all} subgraphs near$p_c$ we suggest that under an appropriate modification of the normalizationconstant 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-sizecorrections in the vicinity of $p_c$ scale with $N$ as $\sim \ln^{-2}N$. Alltogether, our results reveal a previously unknown universality in eigenvaluestatistics of sparse matrices close to the percolation threshold.

Largest eigenvalue statistics of sparse random adjacency matrices