Third workshop on
Large Scale Random Structures

Alessandra Bianchi e Samuele Stivanello

Random walks in random media

We consider a one-dimensional process in random media that generalize a model known in the physical literature as Levy-Lorentz gas. The media is provided by a renewal point process, while the dynamics is obtained as the linear interpolation of a random walk on the point process. We aim to investigate the annealed behavior of this process in the case of inter-distances between points and length of the jumps of the random walk being i.i.d. stable random variables with parameters alpha and beta respectively. If beta =2 (finite variance of the random walk) we establish a functional limit theorem for the process, with scaling limit that depends on the stability parameter alpha of the environment. In particular we show that if alpha in (1,2) the system displays a diffusive behavior, while if alpha in (0,1) the behavior of the motion is super-diffusive with scaling limit related to the so called Kesten-Spitzer process. When beta is less than 2 (infinite variance of the random walk) the scaling behavior of the process is still under investigation. Here we present some partial results and discuss the main open questions together with some heuristics and conjectures.

Massimo Campanino

Ornstein-Zernike behaviour for the correlation functions of the ground state of the quantum Ising model with transverse magnetic field

We study the asymptotic behaviour of the correlation functions of the quantum Ising model with tranverse magnetic field above the critical point in $Z^d$. Using the stochastic representation of the model we show that the exact power law correction to the exponential decay is given by $r^{-d/2}$.
Joint work with M. Gianfelice.

Elisabetta Candellero

Oil and water model on vertex transitive graphs

The oil and water model is an interacting particle system with two types of particles and a dynamics that conserves the number of particles, which belongs to the so-called class of Abelian networks. Widely studied processes in this class are sandpiles models and activated random walks, which are known (at least for some choice of the underlying graph) to undergo an absorbing-state phase transition. In this work we show that the oil and water model is substantially different from such models, as it does not undergo an absorbing-state phase transition and is in the regime of fixation at all densities.
Joint work with A. Stauffer and L. Taggi.

Pietro Caputo

Mixing time of PageRank surfers on sparse random digraphs

Given a directed graph G, a parameter α∈(0,1) and a distribution λ over the vertices of G, the generalised PageRank surf on G with parameters α and λ is the Markov chain on the vertices of G such that at each step with probability α the state is updated with an independent sample from λ, while with probability 1-α the state is moved to a uniformly random vertex in the out-neighbourhood of the current state. The stationary distribution is the so-called PageRank vector. We analyse convergence to stationarity of this Markov chain when G is a large sparse random digraph with given degree sequences, in the limit of vanishing parameter α. We identify three scenarios: when α is much smaller than the inverse of the mixing time of G the relaxation to equilibrium is dominated by the simple random walk and displays a cutoff behaviour; when α is much larger than the inverse of the mixing time of G on the contrary one has pure exponential decay with rate α; when α is comparable to the inverse of the mixing time of G there is a mixed behaviour interpolating between cutoff and exponential decay. This trichotomy is shown to hold uniformly in the starting point and for a large class of distributions λ, including widespread as well as strongly localized measures.
This is joint work with Matteo Quattropani.

Francesca Collet

Dynamical aspects of a Curie-Weiss model of self-organized criticality: moderate fluctuations

Certain large dynamical systems have the tendency to organize themselves into a critical state, without any external intervention. These systems exhibit the phenomenon of self-organized criticality (SOC) that since its introduction has been applied to describe quite a number of natural phenomena (e.g., forest fires, earth-quakes, species evolution). The simplest models exhibiting SOC are obtained by forcing standard critical transitions into a self-organized state. The idea is to start with a model presenting a phase transition and to create a feedback from the configuration to the control parameters in order to converge towards a critical state. In [Cerf, Gorny (2016)], the authors designed a Curie-Weiss model of SOC. They modified the equilibrium distribution associated to the generalized Curie-Weiss model by implementing an automatic control of the inverse temperature that, in the infinite volume limit, drives the system into criticality without tuning any external parameter. We will start by discussing how to approach the problem from a non-equilibrium viewpoint and construct a dynamical Curie-Weiss model of SOC. Then we will characterize path-space moderate fluctuations for the magnetization. Our result shows that, under a peculiar space-time scaling and without tuning any external parameter, the typical behavior of the magnetization is critical.
The talk is based on a joint work with Matthias Gorny (Paris) and Richard C. Kraaij (Delft).

Paolo Dai Pra e Marco Formentin

Mean-field models with multiscale structure

A natural way of going beyond mean-field models consists in considering a population comprised by many communities, each containing many individuals. The interactions among individuals, of mean-field type within a single community, suitably scales when individuals belong to different communities. This gives rise to space-time multiscaling phenomena that are well understood in the case of interacting Wright-Fisher diffusions, leading to rigorous renormalization group arguments. We illustrate some example concerning Ising-type models, giving partial results and open problems.

Alessandra Faggionato

Stochastic homogenization in amorphous media and applications to exclusion processes and random resistor network

We consider random walks on marked simple point processes with symmetric jump rates and unbounded jump range. Examples are given by simple random walks on Delaunay triangulations or Mott variable range hopping We present homogenization results for the associated Markov generators. As a first application, we derive the hydrodynamic limit of the simple exclusion process given by multiple random walks as above, with hard-core interaction. As a second application, we derive a limit theorem for the effective conductivity of some classes of random resistor networks.

Francesca Nardi

Hitting time asymptotics for hard-core interactions on bipartite graphs

We study the metastable behaviour of a stochastic system of particles with hard-core interactions in a high-density regime. Particles sit on the vertices of a bipartite graph. New particles appear subject to a neighbourhood exclusion constraint, while existing particles disappear, all according to independent Poisson clocks. We consider the regime in which the appearance rates are much larger than the disappearance rates, and there is a slight imbalance between the appearance rates on the two parts of the graph. Starting from the configuration in which the weak part is covered with particles, the system takes a long time before it reaches the configuration in which the strong part is covered with particles. We will describe results in collaboration with den Hollander and Taati [3] concerning a sharp asymptotic estimate for the expected transition time and show that the transition time is asymptotically exponentially distributed, and identify the size and shape of the critical droplet representing the bottleneck for the crossover. For various types of bipartite graphs the computations are made explicit. Proofs rely on potential theory for reversible Markov chains, and on isoperimetric results. We compare these results with the ones in [2] where the authors considered the same hard-core model, evolving according to Metropolis dynamics on finite grid graphs and investigated the symptotic behavior of the first hitting time between its two maximum-occupancy configurations (called tunneling time). In particular they show how the order-of-magnitude of this first hitting time depends on the grid sizes and on the boundary conditions by means of a novel combinatorial method. The analysis also proved the asymptotic exponentiality of the scaled hitting time and yields the mixing time of the process in the low-temperature limit as side-result. In order to derive these results, the authors extended the model-independent framework in [1] for first hitting times to allow for a more general initial state and target subset.

1. Manzo F., Nardi F.R., Olivieri E., Scoppola S. (2004)
   On the essential features of metastability: tunnelling time and critical configurations.
   J. Stat. Phys, 115, 591-642.
2. Nardi F.R., Zocca A., Borst S.
   Hitting times asymptotics for hard-core interactions on grids.
   J. Stat. Phys, 162(2), 522-576 (2016).
3. F. den Hollander, F.R. Nardi, S. Taati
   Metastability of hard-core dynamics on bipartite graphs
   Electronic Journal of Probability, Vol. 23, paper no. 97, 1-65, (2018).
4. S. Borst, F. den Hollander, F. R. Nardi, M. Sfragara
   Transition time asymptotics of queue-based activation protocols in random-access networks
   arXiv:1807.05851, arXiv preprint, (2018).

Vittoria Silvestri

Planar aggregation models with subcritical fluctuations

The Hastings-Levitov planar aggregation models describe growing random clusters on the complex plane, built by iterated composition of random conformal maps. A striking feature of these models is that they can be used to define natural off-lattice analogues of several fundamental discrete models, such as Eden or Diffusion Limited Aggregation, by tuning the correlation between the defining maps appropriately. In this talk I will discuss shape theorems and fluctuations of large clusters in the subcritical regime.
Based on joint work with James Norris (Cambridge) and Amanda Turner (Lancaster).

Alexandre Stauffer

Competition in randomly growing processes

We consider a random growth process with competition, which was introduced as a tool to analyze a well-known model of dendritic growth from physics. The process can also be regarded as a model for the spread of fake news in a graph. We will discuss the behavior of this processes, its phase transition and the occurrence of coexistence.
This is based on joint works with Elisabetta Candellero, Tom Finn and Vladas Sidoravicius.