Title: Structural results for the Tree Builder Random Walk

Speaker: Giulio Iacobelli (IM-UFRJ)

Monday, June 17, from 3:30 p.m. to 4:30 p.m. (Rio de Janeiro local time)

This meeting will take place at room C116 - Bloco C - CT – Instituto de Matemática – UFRJ. 

Abstract: 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 \gamma\in (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. 

More complete information about the seminars can be found at

http://www.dme.ufrj.br/?page_id=3481

Sincerely,

Organizers: Giulio Iacobelli and Maria Eulalia Vares

Title: Increasing paths in random temporal graphs

Speaker: Gábor Lugosi (ICREA and Universitat Pompeu Fabra)

Abstract: Motivated by modeling time dependent processes on networks like social interactions and infection spread, we consider a version of the classical Erdős–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 Kamčev.

May 20, from 3:30 p.m. to 4:30 p.m. (Rio de Janeiro local time)

Local: This meeting will take place at room C116 - Bloco C - CT – Instituto de Matemática – UFRJ.

Organizers: Giulio Iacobelli e Maria Eulalia Vares

More complete information about the seminars can be found HERE.

Title: Coin turning and the walk it generates

Speaker: Janos Englander (University of Colorado Boulder)

Online Transmission: https://meet.google.com/haf-zcxs-ckx

Abstract: Given a sequence of numbers $(p_n)_{n\ge 2}$ 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 $p_n$, $n\ge 2$, independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as $n\to\infty$?

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).

All the talks are held in English.

More complete information about the seminars can be found at

http://www.dme.ufrj.br/?page_id=3481

Sincerely,

Organizers: Giulio Iacobelli and Maria Eulalia Vares

Title: "Free energy of two-species systems with applications to colloids"

Speaker: Dimitrios Tsagkarogiannis (Università degli Studi dell'Aquila)

Abstract: 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).

April 29, from 3:30 p.m. to 4:30 p.m. (Rio de Janeiro local time)

Local: This meeting will take place on Google Meet, through the link below:

https://meet.google.com/haf-zcxs-ckx

More complete information about the seminars can be found HERE.

Sincerely,
Organizers: Giulio Iacobelli e Maria Eulalia Vares