Odometer of long-range sandpiles in the torus: mean behaviour and scaling limits

In \cite{Cipriani2016}, the authors proved that with the appropriate rescaling, the odometer of the (nearest neighbours) Divisible Sandpile in the unit torus converges to the bi-Laplacian field. Here, we study $\alpha$-long-range divisible sandpiles similar to those introduced in \cite{Frometa2018}. We obtain that for $\alpha \in (0,2)$, the limiting field is a fractional Gaussian field on the torus. However, for $\alpha \in [2,\infty)$, we recover the bi-Laplacian field. The central tool for our results is a careful study of the spectrum of the fractional Laplacian in the discrete torus. More specifically, we need the rate of divergence of such eigenvalues as we let the side length of the discrete torus goes to infinity. As a side result, we construct the fractional Laplacian built from a long-range random walk. Furthermore, we determine the order of the expected value of the odometer on the finite grid. \end{abstract}Comment: 35 pages, 4 figure

Ergodicity versus non-ergodicity for Probabilistic Cellular Automata on rooted trees

In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCA) on rooted d-regular trees $\mathbb{T}^d$. In a first result we extend the results of [10] on trees, namely we prove that to every stationary measure $\nu$ of the PCA we can associate a space-time Gibbs measure $\mu_{\nu}$ on $\mathbb{Z} \times \mathbb{T}^d$. Under certain assumptions on the dynamics the converse is also true. A second result concerns proving sufficient conditions for ergodicity and non-ergodicity of our PCA on d-ary trees for $d\in \{ 1,2,3\}$ and characterizing the invariant product Bernoulli measures.Comment: 17 page

Synchronization and Spin-Flop Transitions for a Mean-Field XY Model in Random Field

We characterize the phase space for the infinite volume limit of a ferromagnetic mean-field XY model in a random field pointing in one direction with two symmetric values. We determine the stationary solutions and detect possible phase transitions in the interaction strength for fixed random field intensity. We show that at low temperature magnetic ordering appears perpendicularly to the field. The latter situation corresponds to a spin-flop transition

Strongly reinforced P\'olya urns with graph-based competition

We introduce a class of reinforcement models where, at each time step $t$, one first chooses a random subset $A_t$ of colours (independent of the past) from $n$ colours of balls, and then chooses a colour $i$ from this subset with probability proportional to the number of balls of colour $i$ in the urn raised to the power $\alpha>1$. We consider stability of equilibria for such models and establish the existence of phase transitions in a number of examples, including when the colours are the edges of a graph, a context which is a toy model for the formation and reinforcement of neural connections.Comment: 32 pages, 5 figure

Local central limit theorem and potential kernel estimates for a class of symmetric heavy-tailted random variables

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 (Green's function) up to order $\mathcal{O} \left( |x|^{\frac{\alpha-2}{3}+\varepsilon} \right)$ for any $\varepsilon>0$ small enough, when $\alpha \in [1,2)$.Comment: 33 page