Alessandra Faggionato (Roma), Antoine Gloria (Paris & Bruxelles), Frank den Hollander (Leiden), Martin Huesmann (Münster), Sabine Jansen (München), Raphaël Lachièze-Rey (Paris), Giovanni Peccati (Luxembourg)

Alessandra Faggionato (Roma): Random resistor networks on simple point processes and Mott's law. pdf

Random resistor networks are a fundamental tool to analyze transport in disordered media. Popular examples are the resistor network on the $d$-dimensional lattice with random conductances, the resistor network built on the supercritical percolation cluster and the Miller-Abrahams resistor network for conduction in amorphous solids as doped semiconductors. The above networks and many others can be described as resistor networks with nodes given by a simple point process on ${\mathbb R}^d$ and with random conductances, whose randomness is stationarity w.r.t. the action of the group $G$ given by ${\mathbb R}^d$ or ${\mathbb Z}^d$. We discuss the scaling limit of the directional conductivity for this large class of resistor networks in terms of the effective homogenized matrix. Then we push further our analysis of the Miller-Abrahams resistor network, derive Mott's law concerning the anomalous low-temperature decay of the conductivity and discuss how its universality is related to the Poisson point process.

Antoine Gloria (Paris & Bruxelles): On Einstein's effective viscosity formula.

Consider a random suspension of rigid particles (centred at a random stationary point process) in a steady Stokes flow. Since particles are rigid, they hinder the flow, and therefore increase its viscosity on large scales, which we define as the effective viscosity in the sense of stochastic homogenization. In one of his 1905 papers, Einstein proposed an experiment to measure the Avogadro number based on viscosity measurements. One of his ingredients is the so-called Einstein effective viscosity formula, which gives a first-order expansion of the effective viscosity in function of the intensity of the point process in the dilute regime. The aim of this talk is to give a general and robust proof of the Einstein formula, and to extend this result to arbitrarily high order (and in particular justify the Batchelor-Green formula at second order). This requires to define high-order intensities of point processes, that capture finer dilute properties of point processes. The approach involves cluster expansions, combinatorics, elliptic regularity, probability theory, and diagrammatic integration.
This is based on joint work with Mitia Duerinckx.

Frank den Hollander (Leiden): Metastability for the Widom-Rowlinson model with grains of general shape. pdf

We consider the Widom-Rowlinson model on a finite torus in $d$-dimensional Euclidean space with grains of general shape. The energy of a grain configuration is determined by its halo, defined as the union of the grains centred at the locations of a Poisson process. We consider a dynamic version of the model in which grains are randomly created and annihilated as if the outside of the torus were an infinite reservoir with a given chemical potential. We start with the empty torus and are interested in the first time when the torus is fully covered by grains. This crossover mimics the transition from a « gas phase» to a « liquid phase». We focus on the metastable regime where the temperature is low and the chemical potential is supercritical. In order to achieve the transition from empty to full, the system needs to create a sufficiently large droplet, called critical droplet, which triggers the crossover. We identify the size and the shape of this critical droplet, which depend on the solution of an associated isoperimetric inequality.
This is a joint work with R. Kotecký (Prague) and D. Yogeshwaran (Bangalore).

Martin Huesmann (Münster): Fluctuations of the displacement in the optimal matching problem. pdf

The optimal matching problem is one of the classical problems in probability theory. A new linearisation ansatz by Caracciolo et al. led to a new wave of results in recent years. We will explain this linearisation ansatz and show that it allows us to prove that the displacement in the optimal matching problem converges to a Gaussian field which scales as the Gaussian free field.

Sabine Jansen (München): Large deviations and distribution of cracks in a chain of atoms at low temperature. pdf

The talk presents results on the low-temperature asymptotics for a one-dimensional chain of atoms. The latter can be modelled as a one-dimensional Gibbs point process or as a Gibbs measure on sequences of interparticle spacings. As the temperature goes to zero at fixed density, either (1) the measure is well approximated by a Gaussian measure in the vicinity of the periodic energy minimizer, or (2) points fill space by alternating approximately periodic patterns with stretches of empty space (gaps). In case (1) we prove Gaussian limit laws and large deviations principles for both bulk behavior and boundary layers. In case (2) we map the system to an effective gas of defects and analyze in detail the length of empty and approximately crystalline domains as a function of defect energy.
Based on joint works with Wolfgang König, Bernd Schmidt and Florian Theil.

Raphaël Lachièze-Rey (Paris): Percolation of random fields excursions. pdf

We consider homogeneous random fields, i.e. real random functions defined on the Euclidean space. We are in particular interested in the percolation properties of their excursion sets in dimension 2, defined as the (random) sets obtained after thresholding the field at some fixed value which is the parameter. We will first present some results obtained for 1- Gaussian random fields and 2- for shot-noise fields, which can be seen as the Poisson counterparts of Gaussian fields. It turns out that under mild assumptions, as in many parametric models, there is a critical value that separates percolation from non-percolation, and we furthermore observe a sharp phase transition around this value. For symmetric auto-dual models, and in particular for Gaussian excursions, the critical value is trivially 0. For non-symmetric shot-noise excursions, we estimate the critical value at high intensity by approximating the shot-noise field by a Gaussian field through a strong invariance principle.
Joint work with Stephen Muirhead, from Melbourne University.

Giovanni Peccati (Luxembourg): Functional inequalities on the Poisson space, via stopping sets and two-scale stabilization.

I will review some recently established functional inequalities on the Poisson space - based on the use of stopping sets and on the concept of (two-scale) geometric stabilization - that can be roughly regarded as refinements of the usual and second-order Poincar\'e inequalities. I will describe some applications to phase transitions in percolation models, as well as to quantitative central limit theorems for functionals of random geometric graphs.
Joint works with G. Last & D. Yogeshwaran, and with R. Lachièze-Rey & X. Yang.

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