Speaker: Adrián González-Casanova

Title: A random model of Poissonian interacting trajectories inspired by the Lenski experiment

Abstract: Random geometric phenomena emerge as a result of mutations and selection within the Lenski experiment. In this presentation, I will describe a Random Model of Poissonian Interacting Trajectories, which can be derived as a scaling limit from classic models in population genetics. This model exhibits a complex geometric framework, offering insights into the effect of clonal interference on evolutionary dynamics. This talk will be based on joint work with Felix Hermann, Renato dos Santos, Andreas Tobias and Anton Wakolbinger.

Speaker: Alexander Drewitz

Title: (Near-)critical behavior of a strongly correlated percolation model and Hausdorff dimension of the critical clusters

Abstract: For (near-)critical independent Bernoulli percolation, particularly profound results have been obtained in the high-dimensional setting as well as on planar lattices. We consider a strongly correlated percolation model — the level sets of the metric graph Gaussian free field — where significant understanding has also been developed regarding its (near-)critical behavior in intermediate dimensions. We will explain the origin of the model's integrability, and discuss its implications for the associated universality class. A particular focus will be on recent results for the  Hausdorff dimension of the critical connected components.

Speaker: Avelio Sepúlveda

Title: Markov properties on decorated planar maps.

Abstract: Based on joint work with Pablo Araya and Luis Fredes. Planar maps are a natural model for discretizing two-dimensional random geometry, and they have been used to understand the properties of two-dimensional statistical physics models as some of them are more tractable on random planar maps than on the Euclidean lattice Z². A key feature underlying this phenomenon is that these models often satisfy  Markov property. In this talk, I will explore the Markov property on (decorated) planar maps from three perspectives. First, I will describe which probability laws on planar maps satisfy it. Second, I will discuss which random subsets of the map induce a Markovian decomposition. Finally, I will introduce a new probability measure satisfying the Markov property on metric decorated planar maps, a setting where each edge of the map is replaced by a copy of an interval.

Speaker: Eleanor Archer

Title: Limits of uniform spanning trees

Abstract: Uniform spanning trees are a popular model in statistical physics and have many interesting links with other areas of mathematics. In this talk we'll take a look at the large-scale geometry of uniform spanning trees (scaling limits and local limits). Along the way we will see different sampling algorithms and some fundamental links with random walks and electrical networks.

Speaker: Milton Jara

Title: Quantitative hydrodynamics

Abstract: Using the weakly asymmetric exclusion process as an example, we present a new version of the theory of hydrodynamic limits that allows to derive quantitative limits (both law of large numbers and Central Limit Theorems) for the empirical density of particles. Joint with Juan Arroyave, Freddy Hernández, Patricia Gonçalves, Rodrigo Marinho e Otávio Menezes.

Speaker: Natalia Cardona-Tobon

Title: The contact process on dynamical random trees with degree dependence

Abstract: In this talk we investigate the contact process in the case when the underlying structure evolves dynamically as a degree-dependent dynamical percolation model. Starting with a connected locally finite base graph we initially declare edges independently open with a probability that is allowed to depend on the degree of the adjacent vertices and closed otherwise. Edges are independently updated with a rate depending on the degrees and then are again declared open and closed with the same probabilities. We are interested in the contact process, where infections are only allowed to spread via open edges. Our aim is to analyse the impact of the update speed and the probability for edges to be open on the existence of a phase transition. For a general connected locally finite graph, our first result gives sufficient conditions for the critical value for survival to be strictly positive. Furthermore, in the setting of Bienaymé-Galton-Watson trees, we show that the process survives strongly with positive probability for any infection rate if the offspring distribution has a stretched exponential tail with an exponent depending on the percolation probability and the update speed. In particular, if the offspring distribution follows a power law and the connection probability is given by a product kernel  and the update speed exhibits polynomial behaviour, we provide a complete characterisation of the phase transition. This talk is based on joint work with Marcel Ortgiese (University of Bath), Marco Seiler (University of Frankfurt) and Anja Sturm (University of Göttingen).

Speaker: Taísa Lopes Martins

Title: A Lower bound for set-colouring Ramsey numbers

Abstract: In this talk, through the set-colouring Ramsey problem, we illustrate the use of a random construction to obtain an object with a desired property. The set-colouring Ramsey number Rr,s(k) is defined to be the minimum n such that if each edge of the complete graph Kn is assigned a set of s colours from {1, . . . , r}, then one of the colours contains a monochromatic clique of size k. The case s = 1 is the usual r-colour Ramsey number, and the case s = r − 1 was studied by Erdős, Hajnal and Rado in 1965, and by Erdős and Szemerédi in 1972. The first significant results for general s were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraëte, who showed that Rr,s(k) = 2^{Θ(kr)} if s/r is bounded away from 0 and 1. In the range s = r − o(r), however, their upper and lower bounds diverge significantly. In this talk we introduce a new (random) colouring, and use it to determine Rr,s(k) up to polylogarithmic factors in the exponent for essentially all r, s and k. This is a joint work with Lucas Aragão, Maurício Collares, João Pedro Marciano and Rob Morris.

Speaker: Vittoria Silvestri

Title: Branching Internal DLA

Abstract: Internal DLA is a random aggregation process in which the growth of discrete clusters is governed by the harmonic measure seen from an internal point. That is, a simple random walk is released from inside the cluster, and its exit location is added to it. The asymptotic shape of IDLA on Zd starting from a single seed has long been known to be a Euclidean ball, with very small fluctuations. In this talk I will discuss a natural variant of IDLA, namely Branching IDLA, in which the particles that drive the process perform critical branching random walks rather than simple random walks. We will show that BIDLA has a strikingly different phenomenology, namely we prove a phase transition from macroscopic fluctuations in low dimension to the existence of a shape theorem in higher dimension. Based on a joint work with Amine Asselah (Paris-Est Créteil) and Lorenzo Taggi (Rome La Sapienza).