MARK KAC SEMINAR

The divisible sandpile model, a continuous version of the (discrete) abelian sandpile model was introduced by Levine and Peres (2009, 2010) to study scaling limits of the rotor aggregation and internal DLA growth models. The basic mechanism in these models is that to each site of some graph there is associated a height or mass. If the height exceeds a certain value then it collapses by distributing the excess mass (uniformly) to the neighbours which can then result in a series of cascades. In a recent work Levine et al. (2015) addressed two interesting questions regarding these models. One is the dichotomy between stabilizing and exploding configurations and the other the behaviour of the odometer which is a function measuring the amount of mass emitted during stabilization.

They proved that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bi-Laplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bi-Laplacian field.

In this talk we want to introduce some recent results ([1], [2]) in which we show that depending on the initial distribution in any dimension the rescaled odometer converges to the continuum bi-Laplacian field on the unit torus or alpha-stable distribution. Moreover we have some results about stabilization versus explosion for heavy-tailed initial distributions. This is joint work with A. Cipriani (WIAS) and R. Hazra (ISI Kolkata).

1. Alessandra Cipriani, Rajat Subhra Hazra, Wioletta M. Ruszel, Scaling limit of the odometer in divisible sandpiles, arXiv:1604.03754
2. Alessandra Cipriani, Rajat Subhra Hazra, Wioletta M. Ruszel, The divisible sandpile with heavy-tailed variables, arXiv:1610.09863

For the Curie–Weiss (CW) and the random-field Curie–Weiss model a range of limiting results for the empirical magnetisation are known: first of all we have a large deviation principle (LDP), second, we have various (non-standard) central limit theorems (CLTs) where the limiting distribution depends on whether the model-parameters are critical or not, and, thirdly, we have moderate deviation principles (MDP) for the scaling range in between the LDP and CLTs. In the setting where we consider Glauber dynamics for these models, also the LDP and CLTs for the dynamics of the empirical magnetisation are known. For the Curie–Weiss model, we show that we have moderate deviation principles for scalings in between the LDP and CLT. In the setting of the random field CW model, we obtain this result only for a range of scalings in between the LDP and CLT, where the failure for a part of the range is due to fluctuations in the external magnetic field. Finally, we relate the path-space rate functions of the LDP and MDP for the CW model via Gamma convergence, effectively creating a commuting diagram of MDP, Gamma convergence and the LDP.

Based on joint work with Francesca Collet (Delft).