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We prove general nonlinear large deviation estimates similar toChatterjee-Dembo's original bounds except that we do not require any secondorder smoothness. Our approach relies on convex analysis arguments and is validfor a broad class of distributions. Our results are then applied in threedifferent setups. Our first application consists in the mean-fieldapproximation of the partition function of the Ising model under an optimalassumption on the spectra of the adjacency matrices of the sequence of graphs.Next, we apply our general large deviation bound to investigate the largedeviation of the traces of powers of Wigner matrices with sub-Gaussian entries,and the upper tail of cycles counts in sparse Erd\"os-Renyi graphs down to theconnectivity threshold $n^{-1/2}$.

Nonlinear large deviation bounds with applications to traces of Wigner matrices and cycles counts in Erd\"os-Renyi graphs