Seminários de Probabilidade 2024 – 1º Semestre

15/01 (presencial)

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.

Slides Here.

11/03 (presencial)

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).

Slides Here.

25/03 (presencial)

Abstract: Consider dice that are allowed to have different number 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.

Slides Here.

08/04 (presencial)

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.

Slides Here.

29/04 (online)

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).

Link Youtube.

06/05 (presencial)

In financial risk management, modelling dependency within a random vector is crucial and a standard approach is the use of a copula model. A flexible family of copulas, known as the factor copulas, is formed by the copulas extracted from factor models. Sampling from a factor copula is equivalent to sampling from the factor model and applying the cumulative distribution function (c.d.f.) to each component of the sample. Nonetheless, in many models of interest the c.d.f.’s are not explicitly known. In this talk I’ll present theoretical and numerical properties of a transform Markov Chain Monte Carlo (MCMC) scheme developed to efficiently compute expectations conditional to rare events in which the unconditional distribution is given by an intractable factor copula.

Slides Here.

20/05 (presencial)

Motivated by modeling time dependent processes on networks like social interactions
and infection spread, we consider a version of the classical Erdös–Rényi random graph
G(n, p) where each edge has a distinct random time stamp, and connectivity is constrained to
sequences of edges with increasing time stamps. We study the lengths of increasing paths:
the lengths of the shortest and longest paths between typical vertices, the maxima of these
lengths from a given vertex, as well as the maxima between any two vertices; this covers the
(temporal) diameter.
This talk is based on joint work with Nicolas Broutin and Nina Kamöcev.

Slides Here.

03/06 (online)

Given a sequence of numbers $(p_n)_{nge 2}$ in $[0,1]$, consider the following experiment. First, we flip a fair coin and then, at step $n$, we turn the coin over to the other side with probability $p_n$, $nge 2$, independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as $ntoinfty$?
We show that a number of phase transitions take place as the turning gets slower (i.~e.~$p_n$ is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is $p_n=text{const}/n$. Among the scaling limits, we obtain Uniform, Gaussian, Semicircle and Arcsine laws.
If time permits I will also discuss the random walk that coin turning generates. Here each step is 1 or -1 according to what the coin shows. In the unlikely case we have even more time, I will discuss the higher dimensional analogs of the walk.
This is joint work with Stas Volkov (Lund).

Slides Here.

Link Youtube

17/06 (presencial)

The Tree Builder Random Walk (TBRW) is a randomly growing tree built by a walker as it walks around the tree. At each time n, the walker adds a leaf to its current vertex with probability p_n and then moves to a uniform random neighbor on the possibly modified tree. When p_n= n^{-gamma} with gammain (2/3,1], we show that the tree process at its growth times can be coupled to be identical to the Barabási-Albert (BA) preferential attachment model. The coupling also implies that many properties known for the BA-model, such as diameter and degree distribution, can be directly transferred to our TBRW-model.

Slides here

24/06 (presencial)

We consider the (phenomenological) model proposed by Bouchaud and Dean for the dynamics of a (mean field) hierarchical spin glass (following the tree structure proposed by Parisi) at low temperature, and take its limit under different scalings of time and volume, where the limit is either an ergodic process or exhibits aging. Joint work with Andrea Hernández Delgado.

Slides here

15/07 (online)

We prove the existence of an active phase for activated random walks on the lattice in all dimensions. This interacting particle system is made of two kinds of random walkers, or frogs: active and sleeping frogs. Active frogs perform simple random walks, wake up all sleeping frogs on their trajectory and fall asleep at constant rate $lambda$. Sleeping frogs stay where they are up to activation, when woken up by an active frog.
At a large enough density, which is increasing in $lambda$ but always less than one,
such frogs on the torus form a metastable system. We prove that $n$ active frogs in a cramped torus will typically need an exponentially long time to collectively fall asleep
—exponentially long in $n$.
This completes the proof of existence of a non-trivial phase transition for this model designed for the study of self-organized criticality. This is a joint work with Amine Asselah and Nicolas Forien.