Scale-free percolation mixing time

Assign to each vertex of the one-dimensional torus i.i.d. weights with a heavy-tail of index τ−1>0. Connect then each couple of vertices with probability roughly proportional to the product of their weights and that decays polynomially with exponent α>0 in their distance. The resulting graph is called scale-free percolation. The goal of...

Maximum of the Membrane Model on Regular Trees

The discrete membrane model is a Gaussian random interface whose inverse covariance is given by the discrete biharmonic operator on a graph. In literature almost all works have considered the field as indexed over Zd, and this enabled one to study the model using methods from partial differential equations. In this article we would like to...

Dynamical fitness models: evidence of universality classes for preferential attachment graphs

In this paper we define a family of preferential attachment models for random graphs with fitness in the following way: independently for each node, at each time step a random fitness is drawn according to the position of a moving average process with positive increments. We will define two regimes in which our graph reproduces some features...

Scaling Limits in Divisible Sandpiles: A Fourier Multiplier Approach

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)....

Scaling limit of the odometer in divisible sandpiles

In a recent work Levine et al. (Ann Henri Poincaré 17:1677–1711, 2016. https://doi.org/10.1007/s00023-015-0433-x) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the...