Upper tail behavior of the number of triangles in random graphs with constant average degree
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Let N be the number of triangles in an Erdős-Rényi graph 𝒢(n,p) on n vertices with edge density p=d/n, where d>0 is a fixed constant. It is well known that N weakly converges to the Poisson distribution with mean d^3/6 as n→∞. We address the upper tail problem for N, namely, we investigate how fast k must grow, so that the probability of {N≥ k} is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by Poisson behavior only when k^1/3log k< (3/√(2))^2/3log n (sub-critical regime) as well as pin down the tail behavior when k^1/3log k> (3/√(2))^2/3log n (super-critical regime). We further prove a structure theorem, showing that the sub-critical upper tail behavior is dictated by the appearance of almost k vertex-disjoint triangles whereas in the supercritical regime, the excess triangles arise from a clique like structure of size approximately (6k)^1/3. This settles the long-standing upper-tail problem in this case, answering a question of Aldous, complementing a long sequence of works, spanning multiple decades, culminating in (Harel, Moussat, Samotij,'19) which analyzed the problem only in the regime p≫1/n. The proofs rely on several novel graph theoretical results which could have other applications.
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