In this article, we study a class of heavy-tailed random variables on $mathbb{Z}$ in the domain of attraction of an $alpha$-stable random variable of index $alpha in (0,2)$ satisfying a certain expansion of their characteristic function. Our results
include sharp convergence rates for the local (stable) central limit theorem of order $n^{- (1+ frac{1}{alpha})}$, a detailed expansion of the characteristic function of a long-range random walk with transition probability proportional to $|x|^{-(1+alpha)}$ and $alpha in (0,2)$ and furthermore detailed asymptotic estimates of the discrete potential kernel (Greens function) up to order $mathcal{O} left( |x|^{frac{alpha-2}{3}+varepsilon} right)$ for any $varepsilon>0$ small enough, when $alpha in [1,2)$.
We consider the stationary OConnell-Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochas
tic Burgers equation. The proof does not rely on the Cole-Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann-Gibbs principle.
In cite{Cipriani2016}, the authors proved that, with the appropriate rescaling, the odometer of the (nearest neighbours) divisible sandpile on the unit torus converges to a bi-Laplacian field. Here, we study $alpha$-long-range divisible sandpiles, si
milar to those introduced in cite{Frometa2018}. We show that, for $alpha in (0,2)$, the limiting field is a fractional Gaussian field on the torus with parameter $alpha/2$. However, for $alpha in [2,infty)$, we recover the bi-Laplacian field. This provides an alternative construction of fractional Gaussian fields such as the Gaussian Free Field or membrane model using a diffusion based on the generator of Levy walks. The central tool for obtaining our results is a careful study of the spectrum of the fractional Laplacian on the discrete torus. More specifically, we need the rate of divergence of the eigenvalues as we let the side length of the discrete torus go to infinity. As a side result, we obtain precise asymptotics for the eigenvalues of discrete fractional Laplacians. Furthermore, we determine the order of the expected maximum of the discrete fractional Gaussian field with parameter $gamma=min {alpha,2}$ and $alpha in mathbb{R}_+backslash{2}$ on a finite grid.
