The workshop will take place at the Weierstrass Institute (WIAS), which is located in a historic building in the center of Berlin.

Address: Mohrenstraße 39, 10117 Berlin, Germany (Subway station Hausvogteiplatz).

Lecture room: ESH (ground floor)

The workshop will last from Monday 24 June in the morning to approximately the mid-afternoon of Wednesday 26 June. Mornings will be devoted to the minicourses, each one composed by three lectures, 90 minutes each.

On the evening of Monday 24 June there will be the poster session and a social event at Weierstrass Institute.

The program can be found here.

David Dereudre (Université Lille 1): Phase transition for Continuum Gibbs Particle Systems
Bibliography
In this mini-course we give a state of the art of the phase transition phenomenon for continuum interacting particles system. The model is defined in the infinite volume regime via the DLR equations which prescribe the local conditional densities following the Gibbs-Boltzmann formalism. The equilibrium states are also the thermodynamic limit of finite volume volume Gibbs models for any boundary condition. Changing the boundary condition at infinity can change drastically the equilibrium states. We call this phenomenon, phase transition. Two kind of phase transition can appear. The first one, called liquid-gas phase transition, preserves the symmetries of the model (translations, rotations, etc) but the density, the energy or the entropy of particles change abruptly at some values of parameters (in general activity or inverse temperature). The second one, called symmetry breaking phase transition, break some symmetries of the model even though the interaction satisfies these symmetries. Depending on the model, the translation or the rotation invariance is violated. This topic has a long history in Physics and Mathematical-Physics. Several conjectures have been claimed more than fifty years ago and most of them are still open today. Only few results have been proved rigorously due to the lack of tractability of models. Moreover, in the continuum setting, the combinatorial tools, largely used for Gibbs models with bounded spins, is not really efficient here. However, for particular models with well-chosen interactions, it is possible to prove the phase transition phenomena mentioned above. The proofs are rigorous and are based on several tools in Analysis, Probability theory, Geometry. During the mini-course, we will present these results, we will give partially the proofs and we will present also a large collection of conjectures provided beautiful challenges for present and probably future generations.

Kavita Ramanan (Brown University): Scaling limits of Interacting Particle Systems
Models of large stochastic dynamical systems of homogeneous interacting particle systems on a (possible random) graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles arise in a variety of applications, ranging from engineering to physics. These systems are typically too complex to be amenable to exact analysis. We will first introduce several motivating examples of such models, and then describe a range of convergence results that describe the limits of such systems both in the regime of sparse and dense sequences of graphs.

Oriane Blondel (Université Claude Bernard Lyon 1)
Hydrodynamic limit for a facilitated exclusion process
We show a hydrodynamic limit for the exclusion process on ℤ in which a particle can jump to the right only if it has a particle to its left and vice-versa. This process has an active/inactive phase transition at density 1/2. Joint work with Clément Erignoux, Makiko Sasada and Marielle Simon.

Paolo Dai Pra (Università degli Studi di Padova)
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.

Patrik Ferrari (Universität Bonn)
Space-time limit process of KPZ models
We consider stochastic growth models in the Kardar-Parisi-Zhang universality class. The large time limit processes of the interface at fixed time are by now relatively well understood. In the recent few years more efforts have been put in understanding the space-time correlations in the height function. In my talk I will report on recent results in this direction.

Stefan Grosskinsky (University of Warwick)
Dynamics of condensation transitions in stochastic particle systems
We study stochastic particle systems as models of cluster aggregation driven by monomer exchange, and establish the propagation of chaos and a law of large numbers for empirical mass distributions in a mean-field scaling limit under generic growth conditions on particle jump rates. The limiting single-site dynamics of the particle system is a non-linear birth-death chain, and conservation of mass leads to non-uniqueness of stationary measures and a non-trivial ergodic behaviour, which can also involve metastable states and coarsening for condensing particle systems. If growth conditions on the jump rates are violated the system can exhibit finite-time blow up, which we illustrate for an example with product interaction kernel. This is joint work with Watthanan Jatuviriyapornchai and Andre Schlichting.

Benedikt Jahnel (WIAS Berlin)
Attractor properties for irreversible and reversible interacting particle systems
In this talk I will consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a time-stationary measure. The dynamics can be irreversible, which allows us to also treat a class of models exhibiting rotational behavior. I will present und discuss conditions under which weak limit points of any trajectory of translation-invariant measures are Gibbs states for the same specification as the time-stationary measure.

Sabine Jansen (LMU-München)
Cluster expansions for Gibbs point processes
Gibbs point processes form an important class of models in statistical mechanics, stochastic geometry and spatial statistics. A notorious difficulty is that many quantities cannot be computed explicitly; for example, the intensity measure of a Gibbs point process (density) is a highly non-trivial function of the intensity of the underlying Poisson point process (activity). As a partial way out, physicists and mathematical physicists have long worked with perturbation series, called cluster expansions. The talk presents some recent results on cluster expansions for pairwise repulsive interactions and explains connections with generating functions of trees, branching processes, Boolean percolation, and diagrammatic expansions of second-order U-statistics.

Barbara Niethammer (Universität Bonn)
Long-time behaviour in Smoluchowski's coagulation equation
Smoluchowski's classical mean-field model for coagulation is used to describe cluster formation and growth in a large variety of applications. A question of particular relevance is the so-called scaling hypothesis, which suggests that the long-time behaviour is universal and described by self-similar solutions or traveling waves respectively. This issue is well understood for some exactly solvable cases, but in the general case many questions are still completely open. I will give an overview of the results that have been obtained in the last decade and explain why we expect that the scaling hypothesis is not true in general.

Daniel Valesin (University of Groningen)
Spatial Gibbs random graphs
Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with small average graph distance between vertices, but adding an edge comes at a cost measured according to the geometry of the ambient physical space. In most cases, we identify the order of magnitude of the average graph distance as a function of the parameters of the model. As the proofs reveal, hierarchical structures naturally emerge from our simple modeling assumptions. Moreover, a critical regime exhibits an infinite number of phase transitions. Joint work with Jean-Christophe Mourrat (ENS Lyon).

The poster session will take place on Monday 24 June, during the social event at WIAS Institute.

Titles and abstracts of the posters can be found here.

Registration is closed.

If you have any questions, please do not hesitate to contact us: ptps19@wias-berlin.de .