Mod-phi convergence and precise deviations

Valentin Féray, Pierre-Loïc Méliot, Ashkan Nikeghbali

Published 2013-04-10, updated 2015-01-04Version 3

In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of random variables $(X_n)$. The random variables considered can be lattice or non-lattice distributed, and single or multi-dimensional; and one obtains precise estimates of the fluctuations $P[X_n \in t_nB]$, instead of the usual estimates for the rate of exponential decay $log(P[X_n \in t_nB])$. In the special setting of mod-Gaussian convergence, we shall see that our approach allows us to identify the scale at which the central limit theorem ceases to hold and we are able to quantify the breaking of symmetry at this critical scale thanks to the residue or limiting function occurring in mod-$\phi$ convergence. The first sections of the article are devoted to a proof of these abstract results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory (multi-dimensional random walks, random point processes), 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 Erdos-Renyi 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. Although the latter methods can only be applied in the more restrictive setting of mod-Gaussian convergence, the large number as well as the variety of examples which are covered there hint at a universality class for second order fluctuations.