In this talk we consider inhomogeneous Bernoulli bond percolation on the graph GxZ, where G is an infinite connected graph with bounded degree and Z is the set of integers. In 1994, Madras, Schinazi and Schonman showed that there is no percolation in Z^d if the edges are open with a probability of q < 1 if they lie on a fixed axis and with a probability of p < p_c(Z^d) otherwise. Here, we consider a region given by boxes with iid radii centered along the vertical axis 0xZ of GxZ. We allow each edge to be open with a probability of q < 1 if it is inside this region and with a probability of p < p_c(GxZ) otherwise. The goal of the talk is to show that, even if the region is connected, occurrence or not of percolation in this inhomogeneous model depends on how sparse and how large are the boxes placed along the axis. We aim to give sufficient conditions on the moments of the radii as a function of the growth of the graph G for percolation not to occur.
This is a joint work with Rémy Sanchis and Daniel Ungaretti.

We consider the parking process on the grid with a simple occupancy scheme, which is defined as follows. Initially, all the sites in $Lambda_n:={-n,ldots ,n}^d$ are empty. At each step, a site is chosen uniformly at random in $Lambda_n$ and if it and its nearest neighbors are empty, the chosen site is occupied. Once occupied, the site remains so forever. We will discuss the statistical properties of the proportion of occupied sites for this model and for its thermodynamic limit defined on the integer grid $mathbb Z^d$.
This talk is based on ongoing work in collaboration with Alejandro Roldán (UdeA, Colombia), Alexander León (UdeA, Colombia) and Cristian Coletti (UFABC, Brazil).

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Consider dice that are allowed to have different numbers of faces and any number on each face. Die A is said to be better than die B, denoted A ▷ B, if it has a larger probability of winning. This ordering of dice is not transitive: it is possible that A ▷ B ▷ C ▷ A. In this talk we present results on the probability of random dice (with i.i.d. faces) forming an intransitive chain, as the number of faces of each die goes to infinity. We prove a Central Limit Theorem for such dice, combining the method of moments with simple graph theory arguments.

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We shall discuss recent progress related to Ramsey numbers, and the relation with problems in probability. The talk will be based on joint work with Marcelo Campos, Rob Morris and Julian Sahasrabudhe.

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In this talk we consider a system of small and large spheres in a continuous medium all interacting via positive potentials. This could describe colloidal particles (large spheres) within a substrate (small spheres). Alternatively, it could provide an idealized picture of what may happen in a phase transition when the new phase is getting formed (large spheres) but we still have small isolated particles (small spheres). One interesting phenomenon that occurs is that despite the repulsive forces between all particles, when we look at the effective system of only big spheres there is an attractive force between them usually referred to as “depletion attraction”. The question we address in this talk is how to compute the free energy of the system, in particular for the renormalized one, i.e., when we first integrate over the small spheres. We will discuss a sufficient condition for the convergence of the related cluster expansion, which involves the surface of the large spheres rather than their volume (as it would have been the case in a direct application of existing methods to the binary system). This is based on joint works with Sabine Jansen (LMU) as well as with Giuseppe Scola (SISSA) and Xuan Nguyen (NYU-Shanghai).