Program:
08:30 - 09:00
09:00 - 09:45
10:00 - 10:45
10:45 - 11:15
11:15 - 12:00
12:00 - 14:00
14:00 - 14:45

Monday, March 04
Registration/Opening
Teixeira
Hilario
Coffee breakBermolenLunch break
Remenik

15:00 - 15:45
15:45 - 16:15
16:15 - 17:00
17:15 - 18:00

GroismanCoffee breakMorenoRolla
Tuesday, March 05

09:00 - 09:45
10:00 - 10:45
10:45 - 11:15
11:15 - 12:00

Imbuzeiro
CortezCoffee breakJara

12:00 - 14:00
14:00 - 14:45
15:00 - 15:45

Lunch break
Rodriguez
Armendariz

15:45 - 16:15
16:15 - 17:00
17:15 - 18:00

Coffee break
LacoinFerrari

Abstracts:
Ines Armendariz
Title: Potential method for community detection
Abstract: The stochastic block model (SBM) is a random graph model with planted clusters, that is,groups of vertices that share a common label, and such that the connection probability between apair of vertices depends on their label. It is a canonical and widely studied model for community
detection, where one wants to identify the clusters from the observation of a realization of thegraph. In this talk we will review some community detection algorithms and their convergenceproperties, and discuss a quite natural method based on random walks on the graph. We will showthat it converges both weakly and strongly, and that it allows for online identification of thecommunities.
Joint work in progress with Nicolás Agote, Pablo Ferrari and Florencia Leonardi.

Paola Bermolen
Title: Algorithmic Advances in Graph Representation Learning for Dynamic Graphs Abstract: The
extraction of patterns and actionable information from graph data is currently one of the mostimportant topics in machine learning. Different from audio or images, the non-regularity in the data
structure makes prediction or classification tasks very diﬃcult. The goal in learning graphrepresentations is to learn a vector per node(embedding) that can be used in the corresponding
downstream task. Although graph data are ubiquitous in many applications, the use of thesetools is still in its infancy, and traditional techniques persist as the go-to methods for the mostpart.
Our work focuses on spectral-based representations of graphs, such as Random Dot Product
Graphs (RDPGs) and its associated inference method Adjacency Spectral Embedding (ASE). The
state of the art in learning such representations has some limitations: i) it becomes prohibitive formedium/large sized graphs, ii) it does not allowworking with missing data, and iii) the invariant nature
under rotations of the solution makes it diﬃcult to use it for tracking representations in sequencesof graphs.
We bring to bear recent advances in non-convex optimization to address these limitations anddemonstrate their impact on RDPG inference. We advocate first-ordergradient descent methods
to solve the embedding problem better and to cope organically with previous limitations. The
effectiveness of the proposed graph representation learning framework is demonstrated in
reproducible experiments with both synthetic and real network data.
Roberto Cortez
Title: Persistence phenomenon in biological neural network models
Abstract: In a biological neural network, the phenomenon of persistence refers to sustainedneural firing that continues long after the triggering stimulus goes away. It has been observed ina diverse set of brain regions and organisms, and it is considereda mechanism for short-term
storage. In this talk, we introduce a simplified stochastic model for the electric potential of $N$
leaky neurons, whose dynamics is driven by inhomogeneous Poisson processes, accounting forthe intrinsic randomness of the firings. The interactions are sparse: upon firing, the neuron
instantaneously increases the potential of a fixed number of randomly chosen neurons. We study
the large network limit and show that it exhibits a phase transition: the activity of the infinite network
either vanishes in finite time, or it persists forever, depending on the parameters of the firingintensity. This allows us to model persistence: we prove that,although the neural activity of a finite
network eventually vanishes, it may persist for a very long time.
Pablo Ferrari
Title: Continuous Box Ball System
Abstract: The box ball system is a discrete space-time conservative transport cellular automatonwith many conserved quantities called solitons. It was introduced by Takahashi and Satsuma in
1996. We study a space-continuous version, where blocks of consecutive occupied boxes aresubstituted by occupied segments contained in thereal numbers, separated by empty segments.
The walk representation of a continuousball configuration is a zig-zag function, whose dynamics
coincides with Pitman transformation, 1975. We describe an intuitive way to identify solitons andto show that solitons are conserved under the dynamics, using the conservation of ``shortestruns''. We also show that the soliton decomposition of some random zig-zag walks give abidimensional Poisson process, whose evolution is a hierarchical translation of the points. Thisextends discrete space results by Nguyen, Rolla, Wang, Gabrielli and the speaker, 2021 2022. Workin preparation with Inés Armendáriz, Pablo Blanc and Davide Gabrielli.
Pablo Groisman
Title: The Kuramoto Model in Random Geometric Graphs.
Abstract: The Kuramoto model is a nonlinear system of ODEs that represents the behavior of
coupled oscillators. The coupling is determined by a given graph and pushes the system towards
synchronization. An important question is whether there isglobal synchronization (the system
converges to a state in which all the phases coincide from almost every initial condition) or if thesystem supports other patterns.
We will consider the Kuramoto model on random geometric graphs in the d dimensional torus
and prove a scaling limit. The limiting object is given by the heat equation. On the one hand thisshows that the nonlinearities of the system disappear under this scaling and on the other hand,provides evidence that stable equilibria of theKuramoto model on these graphs are, as n → ∞, incorrespondence with those of the heat equation, which are explicit and given by twisted states.
In view of this, we conjecture the existence of twisted stable equilibria with high probability as n→ ∞.
We'll prove this conjecture in dimension d=1.
Marcelo Hilario
Title: Percolation on randomly stretched lattice
Abstract: We revisit a model of percolation on a random lattice first studied by Hoffman [CMP 254,1-22, 2005]. Starting from the usual square lattice, rows and columns are selected uniformly atrandom, and all the edges lying along a selected row or column are removed at once. This gives rise
to a randomly stretched version of the square lattice. Given a realization, independent
bond percolation is performed on top of it. The goal of the present talk is to present a proof for thephase transition for this model. Our proof shares some similarities with the original proof by
Hoffman and with the ideas in Kesten et. al. [EJP 27, 1-49 2022] but also brings some new ideasthat we expect to be useful in new situations. Based on a joint ongoing work with Marcos Sá,
Augusto Teixeira, and Remy Sanchis.
Roberto Imbuzeiro
Title: The top eigenvalue of a random tree
Abstract: Choose a n-vertex labelled tree uniformly at random, with n large. In this talk,we show that,with high probability, the top eigenvalue of the tree equals the square root of the maximum
degree, up to a small additive error. This is similar to what wasrecently shown for a critical ErdösRényi random graph by Hiesmayr and McKensie.
Our proof, however, is quite different, and combines three ingredients. The first one is aseries ofprobabilistic lemmas on typical trees. The second one consists of combinatorial arguments forcounting walks in "nice" trees. The third part of the argument is a way to turn a typical tree intoa nice tree that can only make the top eigenvalue larger.
This is joint work with Louigi Addario-Berry (McGill) and Gabor Lugosi (ICREA/UPF).

Milton Jara
Title: Sharp convergence of mean-field models.
Abstract: By means of two examples, the noisy voter model and the SIS model with demographic
factor, we explain how to derive quantitative estimates on the rate of convergence of observablesof Markov chains. As an application, we show that the lawof the SIS model on the endemic phasecan be well approximated by a Curie-Weiss measure.
Joint work with Yangrui Xiang and Enzo Aljovin.

Hubert Lacoin
Title: Strong disorder and very strong disorder are equivalent for directed polymers (jointwith Stefan Junk)
Abstract: The Directed Polymer in a Random Environment is a statistical mechanics model,
which has been introduced (in dimension 1) as a toy model to study the interfaces of the planar
Ising model with random coupling constants. The model was shortly afterwards generalized tohigher dimensions. In this latter case, rather than an effective interface model, the directed
polymer in a random environment can be thought of as modeling the behavior of a stretchedpolymer in a solution with impurities. The interest in the model model, triggered by its richphenomenology has since then generated a plentiful literature in theoretical physics andmathematics. An important topic for the directed polymer is the so-called localization transition.
This transition can be defined in terms of the asymptotic behavior of the renormalized partition
function of the model. If the finite volume partition function converges to an almost surely positive
limit, we say that weak disorder holds. On the other hand, if it converges to zero almost surely,
we say that strong disorder holds. It has been proved that weak disorder implies that thedistribution of the rescaled polymer converges to standard Brownian motion while somelocalization results have been proved under the strong disorder assumptions. Much stronger
characterizations of disorder-induced localization have been obtained under the stronger
assumption that the partition function converges to zero exponentially fast. This latter regime is
known as the very strong disorder regime. It has long been conjectured that strong and verystrong disorder are equivalent. In this talk we will sketch a proof of this conjecture.
Gregorio Moreno
Title: Diffusions with a singular drift
Abstract: In this talk, we will consider SDEs with a distribution-valued drift. I will report on recent
results on existence and uniqueness, convergence of random walks in vanishing environments,and some examples, including Brox diffusion and random polymers. This is based on joint works
with H. Olivero (U. Valparaíso), N. Perkowski (FU-Berlín) and J. Sepúlveda (UC-Chile).
Daniel Remenik
Title: TASEP with inhomogeneous speeds and memory lengthsAbstract: The totally asymmetric simple exclusion process on the integer lattice is one of thebasic models in the KPZ universality class. Explicit formulas have been derived forthe distributionof the process, which have been used to derive its long-time scaling limit. In this talk I will discuss
some extensions of these formulas to the case when particles jump with different rates, or when
their "memory lengths" are different. Basedon these formulas I will present some asymptotic
results for examples of interest that connect them to processes related to the KPZ fixed point,and I will discuss some conjectures.

Pablo Rodriguez
Title: On the phase transition of two branching processes with selection
Abstract: In this talk we will discuss two special stochastic processes, which may be seen as
branching processes with selection. One of them is the accessibility percolation model on
spherically symmetric trees; the other is a branching random walk with barriers. We will present
the motivation behind its formulation and recent results related to the existence of phasetransition in such processes.

Leonardo Rolla
Title: Quasi-stationary distributions for subcritical population models (Joint work with Pablo
Groisman and Célio Terra.)
Abstract: We consider population models with reproduction (branching process, contact
process, branching random walks). This model has a phase transition in terms of the distributionof the number of children. In the subcritical case, every initial distribution is attracted to theempty configuration and, in the lack of a non-trivial stationary distribution, one studies thequasi-stationary behavior of the system. In this talk we will discuss questions of existence,uniqueness and nonuniqueness of the quasi-stationary distribution, and their relation withspatial aspects of the dynamics.
Augusto Teixeira
Title: CLT for a class of random walks in dynamic random environments
Abstract: This talk will start with a brief overview of some recent developments in the study ofrandom walks in dynamic random environments. Then we will present a new result that provides
a CLT for a family of environments that mix fast, but non-uniformly. The proof of the result isbased on what we call the Random Markov Property. Roughly speaking, this technique involves
the positioning of traps throughout the environment, insuch a way that, when the random walksteps over one of them, it sees an independent future. Finally we will present some open
problems of the area that we believe to be exciting and promising.