| Tuesday 10th June | Wednesday 11th June | ||
| 09h30 - 09h50 | Welcome coffee | ||
| 09h50 - 10h40 | Yvan Velenik | 10h00 - 10h50 | Anja Sturm |
| 10h40 - 11h10 | Break | 10h50 - 11h20 | Break |
| 11h10 - 12h00 | Orphée Collin | 11h20 - 12h10 | Ivailo Hartarsky |
| 12h00- 14h00 | Lunch | 12h10 - 14h00 | Lunch |
| 14h00 - 14h50 | Ariane Carrance | 14h00 - 14h50 | Vlada Limic |
| 14h50 - 15h20 | Break | 14h50 - 15h20 | Break |
| 15h20 - 16h10 | Benedikt Jahnel | 15h20 - 16h10 | Arthur Blanc-Renaudie |
| 16h10 - 16h20 | Mini break | ||
| 16h20 - 17h10 | Dalia Terhesiu | ||
Titles and abstracts
Arthur Blanc-Renaudie, CNRS and Université de Rouen Normandie, France
Scaling limit of critical bond percolation on the torus in high dimensions.
Abstract: In this talk, we will see how to extend the known scaling limits for critical Erdős–Rényi graphs, concerning the sizes of connected components and their geometry, to scaling limits for critical bond percolation on the high-dimensional torus.
Based on a series of ongoing collaborations with Nicolas Broutin, Tom Hutchcroft, and Asaf Nachmias.
Ariane Carrance, University of Vienna, Austria
The Ising model on cubic maps in arbitrary genus: integrability and computability
Abstract: Maps decorated by statistical mechanics models play an important role in 2D quantum gravity, and their study relies on combinatorial tools that are markedly different from those used to study models on fixed lattices. In this talk, I will focus more specifically on the Ising model on cubic maps. I will explain its relation to the integrable Kadomtsev--Petviashvili (KP) hierarchy, and how this can be of use to investigate it further.
This is based on a joint work with Mireille Bousquet-Mélou and Baptiste Louf.
Orphée Collin, Université Paris Cité, France
Two-dimensional simple random walk conditioned on avoiding 0 and two-dimendional random interlacements
Abstract : In this talk, I will present joint works with Francis Comets and Serguei Popov. How can one condition the two-dimensional simple random walk (2D-SRW), which is well known to be recurrent, on not hitting the origin ? This question will be answered by defining rigorously the 2D-SRW conditioned to avoid the origin. I will then discuss some of its surprising properties and recent results concerning its speed of escape towards infinity. I will also present the random interlacements model build upon the conditioned 2D-SRW, which was introduced by the aforementioned coauthors (together with Marina Vachkovskaia) in 2015. I will discuss some properties of this model : FKG inequality, 0-1 law, phase transition and behavior at criticality. During the talk, I will also evoke the continuous counterparts of these processes, namely : the two-dimensional Brownian motion conditioned to avoid the unit ball and the two-dimensional random interlacements. If time allows, I will mention some results on the Brownian random interlacements.
Ivailo Hartarsky, CNRS and Université Claude Bernard Lyon 1, France
Cores on lattices
Abstract: Consider activating the n vertices of a discrete square torus one at a time in uniformly random order. For n large, when does an active 3-core (subgraph with minimum degree at least 3 induced by active vertices) appear and what is its size when it does? We answer these questions, revealing a somewhat surprising explosion, stronger than the mean-field phenomenology, covered by the classical work of Łuczak. The tools involved in the proof come from well-established bootstrap percolation theory. Yet, extending the result to other lattices is much more challenging and is the main focus of our work.
This talk is based primarily on joint work with Lyuben Lichev available at https://arxiv.org/abs/2501.18976, but also related joint works with Hugo Duminil-Copin and with Augusto Teixeira.
Benedikt Jahnel, Technical University of Braunschweig, Germany
Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties
Abstract: In this talk, we consider irreversible translation-invariant interacting particle systems on the d-dimensional hypercubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs with respect to the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure with respect to the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems and is joined work with Jonas Köppl.
Vlada Limic, CNRS and Université de Strasbourg, France
Abstract: The talk will be on encoding the spanning and surplus edges of Erdos-Renyi networks. This encoding was constructed by Josué Corujo and the speaker a couple of years ago. It also served them as a base for studying the augmented multiplicative coalescents in a direct (shorter and simpler) fashion than the original one by Bhamidi, Budhiraja and Wang in 2012.
Anja Sturm, University of Göttingen, Germany
On the contact process on dynamical random graphs with degree dependent dynamics
Abstract: Recently, there has been increasing interest in interacting particle systems on evolving random graphs, respectively in time evolving random environments. In this talk we present results on the contact process in an evolving edge random environment on infinite (random) graphs. The classical contact process models the spread of an infection in a structured population. The structure is given by a graph and the infection is passed on along the edges with a constant rate while recovery from the infection happens spontaneously with rate 1. In an edge random environment the edges of the underlying (random) graph may be dynamically opened and closed to infection.
We first give an overview over recent results. Then, we in particular consider (infinite) Bienaymé-Galton-Watson (BGW) trees as the underlying random graph. Here, we focus on an edge random environment that is given by a dynamical percolation whose opening and closing rates and probabilities are degree dependent. This means that any edges between two vertices are independently updated with rate v and subsequently again declared open (or otherwise closed) with probability p. Here, both v and p depend on the degrees of the adjoining vertices. Our results concern the impact of v and p on the critical infection rate for weak (global) and strong (local) survival of the infection. Specifically, we establish conditions under which the contact process undergoes a phase transition.
For a general connected locally finite graph we provide sufficient conditions for the critical infection rate to be strictly positive. Furthermore, in the setting of BGW trees, we provided conditions on the offspring distribution as well as on v and p so that the process survives strongly with positive probability for all positive values of the infection rate. In particular, if the offspring distribution follows a power law (or has a stretched exponential tail) and the connection probability is given by a product kernel (or a maximum kernel) and the update speed exhibits polynomial behaviour, we provide a quite complete characterisation of the phase transition.
This talk is based on joint work with Natalia Cardona-Tobon, Marcel Ortgiese and Marco Seiler.
Dalia Terhesiu, Leiden University, Netherlands
On some stochastic properties of infinite horizon Lorentz gases.
Abstract: Lorentz gases with infinite horizon provide a rich class of deterministic dynamical systems exhibiting stochastic behavior relevant to nonequilibrium statistical mechanics. In this talk, I will discuss old and new results on stochastic properties of such systems, focusing in particular on models where a point particle moves through a periodic array of convex scatterers and is allowed to travel arbitrarily far without collisions.
The newer results are based on joint works with Françoise Pène, and with Peter Balint. Throughout I will mention open questions that could be of interest for other models in statistical mechanics.
Yvan Velenik, Université de Genève, Suisse
Random walk above a concave obstacle: curvature effects
Abstract: Let f: [−1,1] → ℝ be strongly concave and C3. I'll consider a rather general one-dimensional random walk (Sn; -N ≤ n ≤ N) conditioned on S-N = ⌈Nf(-1)⌉, SN = ⌈Nf(1)⌉ and Sn ≥ N f(n/N) for all -N ≤ n ≤ N. I'll explain how one can show that the typical height of the random walk above this obstacle, Sn - N f(n/N), is of order N1/3. I'll then discuss what happens when one relaxes the assumptions on f. Namely, in the special case f(x)=-|x|p with p in [1,∞), I'll explain that the typical height above 0 is of order N(p-1)/(2p-1). I'll end the talk describing possible applications to the Ising model that were our original motivation for studying this problem.
This is a joint work with Sébastien Ott.
