**Title:** Random walks in dynamic random environments

**Speaker:** **Giulio Iacobelli (IM-UFRJ)**

Our next seminar will be held on Monday, **September 9**,** **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. **There will be no transmission online.

**Abstract: **In this talk we will discuss some recent developments in random walks in dynamic random environments, when the environment is given by a realization of a particle system as the SEP or ZRP.

More complete information about the seminars can be found at

https://www.dme.ufrj.br/?page_id=3583

Sincerely,

Organizers: Giulio Iacobelli and Maria Eulalia Vares

**Título:** CLT for a class of random walks in dynamic random environments

**Palestrante:** Augusto Teixeira (IMPA)

**Data:** 26/08/2024

**Hora:** 15:30h

**Local: **Sala C116 - Instituto de Matemática - UFRJ

**Resumo:** In this talk, we will provide a brief history of some recent developments in the study of random walks in dynamic random environments. Then, we will present a new result that establishes the Central Limit Theorem (CLT) for a family of environments that mix rapidly but not uniformly. We will give an outline of the proof, which follows a very simple idea: showing that the random walk cannot escape from a set of renewal traps. We will conclude the seminar by indicating some interesting directions for the continuation of this study.

This presentation is based on joint work with Julien Allasia, Rangel Baldasso, and Oriane Blondel

**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:** 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:** 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