Upper tail for homomorphism counts in constrained sparse random graphs

Abstract

Consider the upper tail probability that the homomorphism count of a fixed graph H within a large sparse random graph Gn exceeds its expected value by a fixed factor 1+δ. Going beyond the Erdős–Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random dn‐regular graphs (provided H has a regular 2‐core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs H1,…,Hk (extending the known results for k=1), and for inhomogeneous graph ensembles (such as the stochastic block model), we bound the upper tail probability by a variational problem analogous to the one that determines its decay rate in the case of sparse Erdős–Rényi graphs.

Large deviations in random Latin squares

“…For EN 2 , we observe that there are O(n 7 ) ways to specify a pair of intercalates that share two edges, and each such pair is present in G with probability (α/n) 6 . There are…”

“…We remark that as a naïve approach to try to prove Theorem 1.2, we might try to study the independent random hypergraph model mentioned earlier (in which each edge is present with probability 1/n independently), and to condition on the (hopefully not too unlikely) event that our random hypergraph is in fact a Latin square. For example, it is possible to study large deviations in random regular graphs with a related approach [6,26] (although the details are highly nontrivial). However, the property of being a Latin square is extremely restrictive, and there does not seem to be any simple independent model that produces a Latin square property with probability greater than about (1/ √ n) n 2 (which is vanishingly small compared to the large deviation probabilities in Theorem 1.2(a-b)).…”

Upper Tails of Subgraph Counts in Sparse Regular Graphs

“…for all fixed δ > 0 and p ≥ C(δ)n −1 log n. (DeMarco-Kahn actually showed the stronger result that one can take C(δ) = 1 in the earlier expression.) Chatterjee and Dembo [5] managed to improve the large deviation principle so that it applies when p → 0 polynomially with n. Lubetzky and Zhao [14] were able to solve the resulting variational problem in the case of triangles, and were able to use the Chatterjee-Dembo result to show that Pr[T n,p ≥ (1 + δ)E(T n,p )] = exp −(1 + o(1))c(δ)p 2 n 2 log 1 p , where c(δ) = min 1 2 δ 2 3 , 1 3 δ , as long as n − 1 42 log n ≤ p ≪ 1. Bhattacharya, Ganguly, Lubetzky, and Zhao [2] generalized this result and were able to compute the upper tail probability Pr[Hom(K, G(n, p)) ≥ (1 + δ)E(Hom(K, G(n, p)))] up to a factor of 1 + o(1) in the exponent for any fixed graph K. With appropriate bounds on p, they were able to compute a constant c(K, δ) such that Pr[Hom(K, G(n, p)) ≥ (1 + δ)E(Hom(K, G(n, p)))] = exp −(c(K, δ) + o(1))p ∆(K) n 2 log 1 p ,…”

“…Because of this, the answers are often quite different in the G(n, p) and G d n setups. Bhattacharya and Dembo [3] were able to compute the correct log-asymptotic of the probability in the Problem, for graphs K such that the 2-core of K is regular, in a sparse range roughly of the form n 1−ǫ(K) ≪ d ≪ n. (The 2-core of K is given by succesively removing all leaves from K until the minimum degree of K is at least 2, and replacing K by its 2-core does not change the probability in the Problem.) In particular, [3] showed that if the 2-core of K is ∆-regular, then However, [3] left open the question of what happens when the 2-core of K is not regular, only proving that exp −Θ p ∆ n 2 log 1 p (where p = d n and ∆ is the maximum degree of the 2-core of K) is never in fact the correct growth rate.…”

Upper tail large deviations of regular subgraph counts in Erdős-Rényi graphs in the full localized regime

“…Much less is known about the large deviations of subgraph counts in random graph models beyond G(n, p). This has been studied in the context of random d n -regular graphs [4] and random hypergraphs [16] when the (hyper)-graphs are not too sparse. It is worthwhile to investigate whether the ideas of [12] and this paper can be adapted to these problems to treat sparser regimes.…”