Moderate deviations for sums of dependent variables, and the method of cumulants

Abstract: Given a sequence of random variables X_n that converge toward a Gaussian distribution, by looking at the next terms in the asymptotic E[exp(zX_n)] = exp(z^2 / 2) (1+ ...), one can often state a principle of moderate deviations. This happens in particular for sums of dependent random variables, and in this setting, it becomes useful to develop techniques that allow to compute the precise asymptotics of exponential generating series. Thus, we shall present a method of cumulants, which gives new results for the deviations of certain observables in statistical mechanics:

- the number of triangles in a random Erdos-Renyi graph;

- and the magnetization of the one-dimensional Ising model.