Moderator: Alessandra Faggionato (Roma)
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.
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.
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.
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.
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.
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 Poincaré
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
& and D. Yogeshwaran, and with R. Lachièze-Rey and X. Yang.