Upper tails via high moments and entropic stability

Moderate Deviations of Triangle Counts in Sparse Random Graphs

“…Note that we used the bound, 2 k 2 8 √ log n, for the penultimate inequality. This proves the required bound provided C 2 14 .…”

“…The proof of the first identity above goes back to the work of Chatterjee and Dembo [9] and Lubetzky and Zhao [10] who proved this identity for p n −1/42 log n. Then the regime where the first equality holds was gradually extended to p n −1/2 (log n) 2 through the work of Eldan [11], Cook and Dembo [12] and Augeri [13]. Finally, Harel, Mousset and Samotij [14] recently completed the proof of both identities above. They also provided an expression for the value of r(δ, p, n) when p 2 n → c ∈ R.…”

“…From this together with (8)(9)(10)(11)(12)(13) and (8)(9)(10)(11)(12)(13)(14), we deduce the following: in order to prove that (8)(9)(10)(11) happens except with probability at most exp(−1.4b), it suffices to prove that…”

“…where the last inequality comes from the definition of the event E given on (8)(9)(10)(11)(12)(13)(14)(15). We let p := (72C 8 b 2 )/(N 2 K 2 ).…”

Large deviations in random latin squares

“…What is the probability it has fewer than half as many as expected? These types of questions have been intensely studied and are intimately related to the development of many important techniques in graph theory and probability theory; see, for example, the monograph of Chatterjee [15] and the more recent works [1, 4, 6, 16, 28]. Beyond Theorem 1.1, we are able to prove the following near‐optimal bounds on large deviation probabilities for intercalates in random Latin squares.…”