Pierre-Loïc Méliot

Mod-ϕ Convergence

In this paper, we use the framework of mod-φ convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables (Xn)n∈N, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations P[Xn ∈ tnB], instead of the usual estimates for the rate of exponential decay log(P[Xn ∈ tnB]). Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first sections of the article are devoted to a proof of these abstract results and comparisons with existing results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory, number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erdős-Rényi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of “weakly dependent” random variables. The large number as well as the variety of examples hint at a universality class for second order fluctuations. Date: September 7, 2018. 1 ar X iv :1 30 4. 29 34 v4 [ m at h. PR ] 2 3 N ov 2 01 5 2 VALENTIN FÉRAY, PIERRE-LOÏC MÉLIOT, AND ASHKAN NIKEGHBALI CONTENTS

Mod-Gaussian Convergence and Its Applications for Models of Statistical Mechanics

In this paper we complete our understanding of the role played by the limiting (or residue) function in the context of mod-Gaussian convergence. The question about the probabilistic interpretation of such functions was initially raised by Marc Yor. After recalling our recent result which interprets the limiting function as a measure of “breaking of symmetry” in the Gaussian approximation in the framework of general central limit theorems type results, we introduce the framework of L1-mod-Gaussian convergence in which the residue function is obtained as (up to a normalizing factor) the probability density of some sequences of random variables converging in law after a change of probability measure. In particular we recover some celebrated results due to Ellis and Newman on the convergence in law of dependent random variables arising in statistical mechanics. We complete our results by giving an alternative approach to the Stein method to obtain the rate of convergence in the Ellis-Newman convergence theorem and by proving a new local limit theorem. More generally we illustrate our results with simple models from statistical mechanics.

Fluctuations in the case of lattice distributions

If φ is an infinitely divisible distribution, recall that its characteristic function writes uniquely as

Dependency graphs and mod-Gaussian convergence

The theory of dependency graphs is a powerful toolbox to prove asymptotic normality. A dependency graph encodes the dependency structure in a family of random variables: roughly we take a vertex for each variable in the family and connect dependent random variables by edges. The idea is that, if the degrees in a sequence of dependency graphs do not grow too fast, then the corresponding variables behave as if independent and the sum of the corresponding variables is asymptotically normal. Precise normality criteria using dependency graphs have been given by Petrovskaya/Leontovich, Janson, Baldi/Rinott and Mikhailov [PL83, Jan88, BR89, Mik91]. These results are black boxes to prove asymptotic normality of sums of partially dependent variables and can be applied in many different contexts. The original motivation of Petrovskaya and Leontovich comes from the mathematical modelisation of cell populations [PL83]. On the contrary, Janson was interested in random graph theory: dependency graphs are used to prove central limit theorems for some statistics, such as subgraph counts, in G(n, p) [BR89, Jan88, JŁR00]; see also [Pen02] for applications to geometric random graphs. The theory has then found a field of application in geometric probability [AB93, PY05, BV07]. More recently it has been used to prove asymptotic normality of pattern counts in random permutations [Bon10, HJ10]. Dependency graphs also generalise the notion of m-dependence [HR48, Ber73], widely used in statistics [Das08].

Examples with an explicit generating function

The general results of Chapters 3 and 6 can be applied in many contexts, and the main difficulty is then to prove for each case that one has indeed the estimate on the Laplace transform given by Definition 1.1.1. Therefore, the development of techniques to obtain mod-φ estimates is an important part of the work. Such an estimate can sometimes be established from an explicit expression of the Laplace transform (hence of the characteristic function); we give several examples of this kind in Section 7.1. But there also exist numerous techniques to study sequences of random variables without explicit expression for the characteristic function: complex analysis methods in number theory (Section 7.2) and in combinatorics (Section 7.3), localisation of zeros (Chapter 8) and dependency graphs (Chapters 9, 10 and 11) to name a few. These methods are known to yield central limit theorems and we show how they can be adapted to prove mod-convergence. We illustrate each case with one or several example(s).

Random character values from central measures on partitions

Our Theorem 9.1.7 can also be used to study certain models of random integer partitions, which encode the lengths of the longest increasing subsequences in random permutations or random words. Consider a random permutation σ = σ n chosen uniformly among the n! permutations of size n. An increasing subsequence of the permutation σ is a subword σ(i1)σ(i2)… σ(i l ) of the word σ(1)… σ(n), with Denote l n = l(σ n ) the longest length of an increasing subsequence of σ n ; it is a random variable with values in \([\![1,n]\!]\). In the 60s, numerical experiments ([Ula61]) lead Ulam to conjecture the existence of a limit in probability and this was proved later by Hammersley, see [Ham72]. The value of c was then computed by Logan and Shepp and separately by Kerov and Vershik to be equal to 2, cf. [LS77, KV77] for the inequality c ≥ 2, and [KV86] for the converse inequality. More recently, the fluctuations \(n^{-1/6}(\ell_{n} - 2\sqrt{n})\) of the length of the longest increasing subsequence were shown to converge to the Tracy-Widom distribution, see [BDJ99, BOO00, Oko00, Joh01].

Mod-Gaussian convergence from a factorisation of the probability generating function

Let \((X_{n})_{n\in \mathbb{N}}\) be a sequence of bounded random variables with nonnegative integer values, and such that \(\sigma _{n}^{2}:=\mathop{ \mathrm{Var}}\nolimits (X_{n})\) tends to infinity. Denote \(P_{n}(t) = \mathbb{E}[t^{X_{n}}]\) the probability generating function of X n . Each P n (t) is a polynomial in t. Then, it is known that a sufficient condition for X n to be asymptotically Gaussian is that P n (t) has negative real roots (see references below). In this chapter, we prove that if the third cumulant L n 3: = κ(3)(X n ) also tends to infinity with light additional hypotheses, then a suitable renormalised version of X n converges in the mod-Gaussian sense. We then give an application for the number of blocks in a uniform set-partition of [n].

Fluctuations in the non-lattice case

In this chapter, we prove the analogue of Theorems 3.2.2 and 3.3.1 when φ is not lattice-distributed; hence, by Proposition 3.1.2, | eη(iu) | < 1 for any u ≠ 0. In this setting, assuming φ absolutely continuous w.r.t. the Lebesgue measure, there is a formula equivalent to the one given in Lemma 3.2.1, namely,

An extended deviation result from bounds on cumulants

In this chapter, we discuss a particular case of mod-Gaussian variables that arises from bounds on cumulants. We will see that in this case the deviation result given in Theorem 4.2.1 is still valid at a scale larger than t n ; see Proposition 5.2.1.

Subgraph count statistics in Erdős-Rényi random graphs

In this chapter, we consider Erdős-Renyi model Γ(n, p n ) of random graphs. A random graph Γ with this distribution is described as follows. Its vertex set is [n] and for each pair {i, j} ⊂ [n] with i ≠ j, there is an edge between i and j with probability p n . Moreover, all these events are independent. We are then interested in the following random variables, called subgraph count statistics. If γ is a fixed graph of size k, then X γ (n) is the number of copies of γ contained in the graph Γ(n, p n ) (a more formal definition is given in the next paragraph). This is a classical parameter in random graph theory; see, e.g. the book of S. Janson, T. Łuczak and A. Rucinski [JŁR00].