Upper tail of the spectral radius of sparse Erd\H{o}s-R\'{e}nyi graphs

We consider an Erd\H{o}s-R\'{e}nyi graph $\mathbb{G}(n,p)$ on $n$ vertices with edge probability $p$ such that \[ \sqrt{\frac{\log n}{\log \log n}} \ll np \le n^{1/2-o(1)}, \] and derive the upper tail large deviations of the largest eigenvalue of the adjacency matrix. Within this regime, we show that for $p \gg n^{-2/3}$ the $\log$-probability of the upper tail of the largest eigenvalue equals to that of planting a clique of an appropriate size (upon ignoring smaller order terms), while for $p \ll n^{-2/3}$ the same is given by that of the existence of a high degree vertex. In the regime, $\log(np) \gtrsim \log n$ the large deviations of the spectral radius is derived from that of homomorphism counts of cycles of even length. A similar dichotomy, as above, arises for the homomorphism counts of cycles of length $2t$, with the {\em threshold depending on $t$}. This, in particular, identifies a sparsity regime, both in the context of the largest eigenvalue and homomorphism counts of even cycles, where the large deviation is due to the presence of a non-planted localized structure. It further demonstrates that in a certain sparsity regime the large deviation rate functions for the homomorphism and subgraph counts differ from each other, while the large deviation speeds continue to be the same, even when the expected value of the former is $1+o(1)$ times that of the latter. Our results also identify the typical behavior of $\mathbb{G}(n,p)$ conditioned on the upper tail events of the spectral radius and homomorphism counts of even cycles in certain sub-regimes of $p$.