In this paper we investigate scaling limits of the odometer in divisible sandpiles on d-dimensional tori following up the works of Chiarini et al. (Odometer of long-range sandpiles in the torus: mean behaviour and scaling limits, 2018), Cipriani et al. (Probab Theory Relat Fields 172:829–868, 2017; Stoch Process Appl 128(9):3054–3081, 2018). Relaxing the assumption of independence of the weights of the divisible sandpile, we generate generalized Gaussian fields in the limit by specifying the Fourier multiplier of their covariance kernel. In particular, using a Fourier multiplier approach, we can recover fractional Gaussian fields of the form (- Δ) ^{- s / 2}W for s> 2 and W a spatial white noise on the d-dimensional unit torus.

In this note we study metastability phenomena for a class of long-range Ising models in one-dimension. We prove that, under suitable general conditions, the configuration - 1 is the only metastable state and we estimate the mean exit time. Moreover, we illustrate the theory with two examples (exponentially and polynomially decaying interaction) and we show that the critical droplet might be macroscopic or mesoscopic, according to the value of the external magnetic field.

We show that in Abelian sandpiles on infinite Galton–Watson trees, the probability that the total avalanche has more than t topplings decays as t^{- 1 / 2}. We prove both quenched and annealed bounds, under suitable moment conditions. Our proofs are based on an analysis of the conductance martingale of Morris (Probab Theory Relat Fields 125:259–265, 2003), that was previously used by Lyons et al. (Electron J Probab 13(58):1702–1725, 2008) to study uniform spanning forests on Z^d, d≥ 3 , and other transient graphs.

We consider ferromagnetic long-range Ising models which display phase transitions. They are one-dimensional Ising ferromagnets, in which the interaction is given by Jx,y=J(|x-y|)≡1|x-y|2-α with α∈ [0 , 1) , in particular, J(1) = 1. For this class of models, one way in which one can prove the phase transition is via a kind of Peierls contour argument, using the adaptation of the Fröhlich–Spencer contours for α≠ 0 , proposed by Cassandro, Ferrari, Merola and Presutti. As proved by Fröhlich and Spencer for α= 0 and conjectured by Cassandro et al for the region they could treat, α∈ (0 , α₊) for α₊= log (3) / log (2) - 1 , although in the literature dealing with contour methods for these models it is generally assumed that J(1) ≫ 1 , we will show that this condition can be removed in the contour analysis. In addition, combining our theorem with a recent result of Littin and Picco we prove the persistence of the contour proof of the phase transition for any α∈ [0 , 1). Moreover, we show that when we add a magnetic field decaying to zero, given by h_x= h_∗· (1 +

We consider the two-dimensional Ising model with long-range pair interactions of the form (Formula presented.) with (Formula presented.), mostly when (Formula presented.). We show that Dobrushin states (i.e. extremal non-translation-invariant Gibbs states selected by mixed ± boundary conditions) do not exist. We discuss possible extensions of this result in the direction of the Aizenman–Higuchi theorem, or concerning fluctuations of interfaces. We also mention the existence of rigid interfaces in two long-range anisotropic contexts.