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 this work is to study the mixing time of the simple random walk on this structure. We depict a rich phase diagram in α and τ. In particular we prove that the presence of hubs can speed up the mixing of the chain. We use different techniques for each phase, the most interesting of which is a bootstrap procedure to reduce the model from a phase where the degrees have bounded averages to a setting with unbounded averages.

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 investigate the dependence of the membrane model on a different geometry, namely trees. The covariance is expressed via a random walk representation which was first determined by Vanderbei in (Ann Probab 12:311–314, 1984). We exploit this representation on m-regular trees and show that the infinite volume limit on the infinite tree exists when m≥ 3. Further we determine the behavior of the maximum under the infinite and finite volume measures.

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 of two well-known preferential attachment models: the Bianconi-Barabási and Barabási-Albert models. We will discuss a few conjectures on these models, including the convergence of the degree sequence and the appearance of Bose-Einstein condensation in the network when the drift of the fitness process has order comparable to the graph size.

In this article we study the scaling limit of the interface model on Zd where the Hamiltonian is given by a mixed gradient and Laplacian interaction. We show that in any dimension the scaling limit is given by the Gaussian free field. We discuss the appropriate spaces in which the convergence takes place. While in infinite volume the proof is based on Fourier analytic methods, in finite volume we rely on some discrete PDE techniques involving finite-difference approximation of elliptic boundary value problems.

We consider a semiflexible polymer in Zd which is a random interface model with a mixed gradient and Laplacian interaction. The strength of the two operators is governed by two parameters called lateral tension and bending rigidity, which might depend on the size of the graph. In this article we show a phase transition in the scaling limit according to the strength of these parameters: we prove that the scaling limit is, respectively, the Gaussian free field, a “mixed” random distribution and the continuum membrane model in three different regimes.

In this article we define the discrete Gaussian free field (DGFF) on a compact manifold. Since there is no canonical grid approximation of a manifold, we construct a random graph that suitably replaces the square lattice Z^d in Euclidean space, and prove that the scaling limit of the DGFF is given by the manifold continuum Gaussian free field (GFF). Furthermore using Voronoi tessellations we can interpret the DGFF as element of a Sobolev space and show convergence to the GFF in law with respect to the strong Sobolev topology.

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.