Speaker: Daniel de la Riva Massaad (UBC)

Title: Sharpness of the Phase Transition for the Frog Model on Transitive Graphs

Abstract: We consider a small modification of the frog model. For a given vertex-transitive graph, each vertex has $Poisson(\lambda)$ particles (or frogs). At time zero, only the particles at the origin are active, and all the other particles are sleeping. Each active particle performs an independent continuous time simple random walk, becoming inactive after time $t$. Once an active frog jumps to a vertex, it activates all of its particles. Unlike previous works, we study the survival of active particles as a dependent percolation model with two parameters $\lambda$ and $t$, establishing the existence of phase transition for certain classes of graphs between the process dying almost surely in finite time and surviving with positive probability. In this talk, we prove sharpness of the phase transition with respect to each of the parameters. Based on a joint work with Omer Angel, Jonathan Hermon, and Yuliang Shi.

Speaker: Jhon Kevin Astoquillca Aguilar (IME-USP)

Title: On the stationary measures of two variants of the voter model 

Abstract: The voter model is a Markov process in which the vertices of a graph (interpreted as voters) adopt one of two opinions (0 and 1), and update their opinions at random times by copying the opinion of a uniformly chosen neighbor. This process is dual to a system of coalescing random walks (if a pair of random walks in this system collide, they continue together for the remainder of their trajectory). The duality implies that the set of stationary measures of the voter model on a graph is related to the collision dynamics of random walks on that graph. In this work, we provide a characterization of the space of stationary measures of the voter model based on the notions of finite and infinite collision, as introduced in [1]. We then present several examples that highlight the necessity of certain assumptions on the underlying graphs. Finally, we extend this result to two variations of the voter model: first, a version that incorporates opinion swapping among voters, and second, the voter model on a graph that coevolves with the process—that is, a dynamic random graph, namely, the dynamical percolation graph. To obtain this extension, we analyze the collision properties of random walks in two contexts: first, with a behavior of position swapping among the random walks, which complicates their collisions; and second, with random walks defined in a dynamical percolation environment. If time permits, we also present some geometric properties of the stationary measures of the voter model in both variations, focusing primarily on the existence of an infinite connected set of voters holding opinion 1 in the extremal stationary measures. This work is based on the author's doctoral thesis [2].

[1] Barlow, Martin T.; Peres, Yuval; Sousi, Perla. Collisions of random walks, AIHP 48 (2012) no. 4.

[2] Astoquillca, Jhon. Synchronization in random networks. [Thesis]. University of Groningen. (2025).

Speaker: João Maia (Peking University)

Title: Phase transitions for the long-range Random Field Ising model

Abstract: The long-range random field Ising model combines two well known variations of the Ising model. The long-range Ising model, sometimes called Dyson model, allows every pair of spins to interact, with interaction strength decaying polynomially in the distance between them. In one dimension, this model exhibits a phase transition in the temperature, different from the nearest-neighbor counterpart. Another variation of the Ising model, called the random field Ising model (RFIM), incorporates an independent Gaussian random external field acting on each site. This model has garnered significant interest recently, and it is most notorious for its lack of a phase transition in two dimensions. In this talk, I will present recent results on the existence and nature of phase transitions in the long-range RFIM. Joint work with L. Affonso, R. Bissacot, J. Ding, and F. Huang.

Speaker: Melissa González García (Paris-Dauphine)

Title: Percolation Games: Zero-sum games on $\mathbb{Z}^d$ with random payoffs

Abstract: Imagine two players moving a token through the vertices of the $d$-dimensional lattice. Each time they make a move, they either win or lose something, but this payoff is random. The value of the game reflects the payoff obtained when both players act rationally. In percolation games, this process is repeated over and over again, generating a sequence of game values. A central question is whether this sequence converges to a limit value as the number of steps approaches infinity. If the game is oriented -meaning that the token moves in a fixed direction irrespective of the players' actions- and the payoffs are independently and identically distributed across space, then the existence of a limit value has been proven. What role does the orientation assumption play? What other features of the game can we focus on to study the long-run behavior of it? These games fall within the broad category of stochastic games and are particularly interesting. They connect with classical percolation theory problems and also offer a discrete-time game-theoretic framework for studying a significant problem in partial differential equations which ties into continuous-time games. In this presentation, we will introduce this class of games and briefly explore a range of examples in different settings. These examples will underscore the challenges and mathematical depth of the model, while also revealing its connections, especially to percolation theory.

Speaker: Sebastian Angel Zaninovich (UBA)

Title: GUE fluctuations near the axis in One-Sided Ballistic Deposition

Abstract: We introduce a variation of the classic ballistic deposition model in which vertically falling blocks can only stick to the top or the upper right corner of growing columns. We establish that the fluctuations of the height function at points near the -axis are given by the GUE ensemble and its correspondent Tracy-Widom limiting distribution. The proof is based on a graphical construction of the process in terms of a directed Last Passage Percolation model.