**Title:** Too many frogs cannot fall asleep

**Speaker:** Alex Gaudillière (Aix-Marseille Université)

Monday, **July 15**,** **from **3:30 p.m. to 4:30 p.m**. (Rio de Janeiro local time)

**Online Transmission:** https://meet.google.com/haf-

**Abstract: **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.

**Title:** Scaling Limits of the Bouchaud and Dean Trap Model on Parisi's Tree

**Speaker:** Luiz Renato Fontes (IME-USP)

Monday, **June 24**,** **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:** 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.

More complete information about the seminars can be found at

**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:** 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**.