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For a $\Delta$-regular connected graph ${\sf H}$ the problem of determiningthe upper tail large deviation for the number of copies of ${\sf H}$ in$\mathbb{G}(n,p)$, an Erd\H{o}s-R\'{e}nyi graph on $n$ vertices with edgeprobability $p$, has generated significant interests. For $p\ll 1$ and$np^{\Delta/2} \gg (\log n)^{1/(v_{\sf H}-2)}$, where $v_{\sf H}$ is the numberof vertices in ${\sf H}$, the upper tail large deviation event is believed tooccur due to the presence of localized structures. In this regime the largedeviation event that the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$exceeds its expectation by a constant factor is predicted to hold at a speed$n^2 p^{\Delta} \log (1/p)$ and the rate function is conjectured to be given bythe solution of a mean-field variational problem. After a series ofdevelopments in recent years, covering progressively broader ranges of $p$, theupper tail large deviations for cliques of fixed size was proved by Harel,Mousset, and Samotij \cite{hms} in the entire localized regime. This paperestablishes the conjecture for all connected regular graphs in the wholelocalized regime.

Upper tail large deviations of regular subgraph counts in Erd\H{o}s-R\'{e}nyi graphs in the full localized regime