Central limit theorems and the geometry of polynomials

Abstract

Let X ∈ {0, . . . , n} be a random variable, with mean µ and standard deviation σ and letbe its probability generating function. Pemantle conjectured that if σ is large and f X has no roots close to 1 ∈ C then X must be approximately normal. We completely resolve this conjecture in the following strong quantitative form, obtaining sharp bounds. If δ = min ζ |ζ − 1| over the complex roots ζ of f X , andwhere Z ∼ N (0, 1) is a standard normal. This gives the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. We also show that if f X has no roots with small argument, then X must be approximately normal, again in a sharp quantitative form:Using this result, we answer a question of Ghosh, Liggett and Pemantle by proving a sharp multivariate central limit theorem for random variables with real-stable probability generating functions.

Absence of zeros implies strong spatial mixing

“…Another application of absence of complex zeros is found in [22,24,30,31], where central limit theorems are derived for discrete probability distributions taking a finite number of values in the nonnegative integers, whose probability generating function p(X), defined as…”

Approximately counting independent sets of a given size in bounded-degree graphs

“…The probability generating function of a discrete random variable X distributed on the non-negative integers is the polynomial in z given by f (z) = j≥0 P(X = j)z j , and the above result shows that at subcritical fugacity the probability generating function of |I| has no zeros close to 1 in C. This lets us use the following result of Michelen and Sahasrabudhe [22].…”

“…A pleasant feature of our methods is the incorporation of several advances from recent research on related topics. From the geometry of polynomials we use a state-of-the-art zero-free region for Z G (λ) due to Peters and Regts [25] and a central limit theorem of Michelen and Sahasrabudhe [23,22] (though an older result of Lebowitz, Pittel, Ruelle and Speer [21] would also suffice), and we also apply the very recent development that a natural Markov chain for sampling from the hard-core model at subcritical fugacities (the Glauber dynamics) mixes rapidly [1,6]. Finally, our results also show a connection between these algorithmic and complexity-theoretic problems and extremal combinatorics problems for bounded-degree graphs [9,12,10], see also the survey [31].…”

Untitled

“…While we ultimately resolved the conjecture of Pemantle by different means (see [33]), Conjecture 1 remains of independent interest and motivates our work here. What is perhaps surprising about this conjecture is that it says that random variables X of this type have variance that is essentially as large as possible, as it is not hard to see that Var(X) = O R,δ (n) .…”