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- Department of Mathematics, MIT

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### ON THE LOWER TAIL VARIATIONAL PROBLEM FOR RANDOM GRAPHS

"... Abstract. We study the lower tail large deviation problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős–Rényi random graph G(n, p). We are interested in estimating the lower tail probability P(XH ≤ (1 − δ)EXH) for fixed 0 < δ < 1. Thanks to the re ..."

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Abstract. We study the lower tail large deviation problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős–Rényi random graph G(n, p). We are interested in estimating the lower tail probability P(XH ≤ (1 − δ)EXH) for fixed 0 < δ < 1. Thanks to the results of Chatterjee, Dembo, and Varadhan, this large deviation problem has been reduced to a natural variational problem over graphons, at least for p ≥ n−αH (and conjecturally for a larger range of p). We study this variational problem and provide a partial characterization of the so-called “replica symmetric ” phase. Informally, our main result says that for every H, and 0 < δ < δH for some δH> 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős–Rényi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite H and δ close to 1. 1. Background We consider large deviations of subgraph counts in Erdős–Rényi random graphs. Fix a graph H, and let XH denote the number of copies of H in an Erdős–Rényi random graph G(n, p). For a fixed δ> 0, the problem is to estimate the probabilities (upper tail) P(XH ≥ (1 + δ)EXH) and