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In their breakthrough 2011 paper, Chatterjee and Varadhan established a large deviations principle (LDP) for the Erdös-Rényi graph G(N,p), viewed as a measure on the space of graphons with the cut metric topology. This yields LDPs for subgraph counts, such as the number of triangles in G(N,p), as these are continuous functions of graphons. However, as with other applications of graphon theory, the LDP is only useful for dense graphs, with p ϵ (0,1) fixed independent of N.

Since then, the effort to extend the LDP to sparse graphs with p ~ N^{-c} for some fixed c>0 has spurred rapid developments in the theory of "nonlinear large deviations". We will report on recent results increasing the sparsity range for the LDP, in particular allowing c as large as 1/2 for cycle counts, improving on previous results of Chatterjee-Dembo and Eldan. These come as applications of new quantitative versions of the classic regularity and counting lemmas from extremal graph theory, optimized for sparse random graphs. (Joint work with Amir Dembo.)