Local limit theorems for subgraph counts

Abstract

We introduce a general framework for studying anticoncentration and local limit theorems for random variables, including graph statistics. Our methods involve an interplay between Fourier analysis, decoupling, hypercontractivity of Boolean functions, and transference between 'fixed-size' and 'independent' models. We also adapt a notion of 'graph factors' due to Janson. As a consequence, we derive a local central limit theorem for connected subgraph counts in the Erdős-Renyi random graph 𝐺(𝑛, 𝑝), building on work of Gilmer and Kopparty and of Berkowitz. These results improve an anticoncentration result of Fox, Kwan, and Sauermann and partially answer a question of Fox, Kwan, and Sauermann. We also derive a local central limit theorem for induced subgraph counts, as long as 𝑝 is bounded away from a set of 'problematic' densities, partially answering a question of Fox, Kwan, and Sauermann. We then prove these restrictions are necessary by exhibiting a disconnected graph for which anticoncentration for subgraph counts at the optimal scale fails for all constant 𝑝, and finding a graph 𝐻 for which anticoncentration for induced subgraph counts fails in 𝐺(𝑛, 1∕2). These counterexamples resolve anticoncentration conjectures of Fox, Kwan, and Sauermann in the negative. Finally, we also examine the behavior of counts of 𝑘-term arithmetic progressions in subsets of ℤ∕𝑛ℤ and deduce a local limit theorem wherein the behavior is Gaussian at a global

Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture

“…If this strategy were to succeed, it would reveal that in this case the true distribution of X is Gaussian on two different scales: When ‘zoomed out’, we see a bell curve with standard deviation about , but ‘zooming in’ reveals a superposition of many smaller bell curves each with standard deviation about n (see Figure 1). This kind of behavior can be described in terms of a so-called Jacobi theta function and has been observed in combinatorial settings before (by the second and fourth authors [88]).…”

“…To exploit cancellation from the linear part, we adapt a decorrelation technique first introduced by Berkowitz [12] to study clique counts in random graphs (see also [88]), involving a subsampling argument and a Taylor expansion. While all previous applications of this technique exploited the particular symmetries and combinatorial structure of a specific polynomial of interest, here we instead take advantage of the robustness inherent in the definition of RLCD.…”

Approximate counting and sampling via local central limit theorems

“…For our proof of the LCLT for the hard-core model, we will need the following simple lemma. The precise version we state here appears in work of Berkowitz [6] on quantitative local central limit theorems for triangle counts in G(n, p) and has been used, for instance, in further work on local central limit theorems for general subgraph counts in random graphs [38]. Lemma 2.5 ([6, Lemma 3]).…”

Approximate counting and sampling via local central limit theorems

“…For our proof of the LCLT for the hard-core model, we will need the following simple lemma. The precise version we state here appears in work of Berkowitz [6] on quantitative local central limit theorems for triangle counts in G(n, p) and has been used, for instance, in further work on local central limit theorems for general subgraph counts in random graphs [37]; we include the short proof for the reader's convenience. Lemma 2.5 ([6, Lemma 3]).…”