# Unraveling parking on random trees via random graphs

### Date: Oct 14, 2020 at 16:15

Abstract: Imagine a plane tree together with a configuration of particles (cars) at each vertex.Each car tries to park on its node, and if the latter is occupied, it moves downward towards the root trying to find an empty slot.When the underlying plane tree is a critical Galton–Watson conditioned to be large, and when the cars arrivals are i.i.d. on each vertex, we observe a phase transition:- when the density of cars is small enough, all but a few manage to park safely,- whereas when the density of cars is high enough, a positive fraction of them do not manage to park and exit through the root of the tree.The critical density is an explicit function of the first two moments of the offspring distribution and cars arrivals (C. & Hénard 2019).We shall give a new point of view on this process by coupling it with the ubiquitous Erdös–Rényi random graph process.This enables us to fully understand the (dynamical) phase transition in the scaling limit by relating it to the multiplicative coalescent process. The talk is based on a joint work with Olivier Hénard and an ongoing project with Alice Contat.

Venue: Online via Zoom (if you are interested in the link please send a message to the organizers through the contact form)

# Facilitated Exclusion Process and Stefan Problem

### Date: Sep 23, 2020 at 16:00

Abstract: We consider an exclusion-type particle process on Z, in which a particle can only jump if it is “pushed” by another particle. We show that the hydrodynamic limit is given by a Stefan problem and that the active/inactive phases at the micro- and macroscopic levels coincide.

Joint work with Clément Erignoux and Marielle Simon.

Venue: Online via Zoom (if you are interested in the link please send a message to the organizers through the contact form)

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# A mean-field theory for certain deep neural networks

### Date: Aug 20, 2020 at 15:00

Abstract: A natural approach to understand overparameterized deep neural networks is to ask if there is some kind of natural limiting behavior when the number of neurons diverges. We present a rigorous limit result of this kind for networks with complete connections and “random-feature-style” first and last layers. Specifically, we show that network weights are approximated by certain “ideal particles” whose distribution and dependencies are described by McKean-Vlasov mean-field model. We will present the intuition behind our approach; sketch some of the key technical challenges along the way; and connect our results to some of the recent literature on the topic.

Venue: Online via Zoom (if you are interested in the link please send a message to the organizers through the contact form)

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# Sobre el fenómeno de transición de fase en percolación

### Date: Jul 2, 2020 at 14:30

Abstract: Consideremos una subdivisión del plano en hexágonos como una colmena, y coloreemos cada célula en negro con probabilidad p de manera independiente. La cantidad de células negras crece linealmente con p. Sin embargo, al estudiar en especial propiedades de conexiones a gran escala, se observan fenómenos de umbral, centrados alredededor el parámetro $$p_c=1/2$$. Se trata pues de una transición de fase. Este fenómeno es descrito por el teorema de Harris-Kesten demostrado en parte en 1960, y completado en 1980.
Desde entonces, en mecánica estadística, se han estudiado muchos otros modelos, como el modelo de Ising, la percolación de Voronoi, percolaciones dependientes etc. que también inducen coloraciones aleatorias del plano, y en las cuales también se observan fenómenos de transición de fase. Esto ha llevado los probabilistas a desarrollar diferentes técnicas para generalizar el teorema de Harris-Kesten y llegar a un mejor entendimiento de tales fenómenos.
En una colaboración reciente con Hugo Vanneuville y Stephen Muirhead, proponemos un avance en esta historia. Demostramos una transición de fase para una clase de modelos construidos a partir de campos Gaussianos. En esta charla quisiera presentar las principales estrategias empleadas y la especificidad de nuestra técnica.

Venue: Online via Zoom
Coordinators: Prof. Joaquín Fontbona & Prof. Daniel Remenik

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# Random time transformation analysis of Covid19 2020

### Date: Jun 18, 2020 at 14:30

Abstract: The SIR epidemiological equations model new affected and removed cases as roughly proportional to the current number of infected cases. An alternative that has been considered in the literature will be adopted, in which the number of new affected cases is proportional to the $$\alpha$$ power of the number of infected cases. After arguing that $$\alpha = 1$$ models exponential growth while $$\alpha <1$$ models polynomial growth, a simple method for parameter estimation in differential equations subject to noise, the random-time transformation RTT of Bassan, Meilijson, Marcus and Talpaz 1997, will be reviewed and applied in an attempt to settle the question as to the nature of Covid19.

Venue: Online via Zoom
Coordinators: Prof. Joaquín Fontbona & Prof. Daniel Remenik

# Topological phase transition as a statistical reconstruction problem

### Date: Jun 04, 2020 at 14:30 h

Abstract: Joint work with C. Garban. KT or topological phase transitions are a type of  phase transition discovered by Kosterlitz and Thouless in the ’70s. Models that undergo this phenomenon are typically 2-dimensional and do not have a classical phase transition. In this talk, I will explain this type of phase transition going over the first proof of their existence by Fröhlich and Spencer which relates them to the localization of random surfaces. Then, I will discuss a new interpretation of this phase transition that arises from the following question: Let $$\phi$$ be a discrete Gaussian free field at temperature $$T$$ and imagine that you lost its integer part, can you recover the macroscopic information of $$\phi$$?.

Venue: Online via Zoom
Coordinators: Prof. Joaquín Fontbona & Prof. Daniel Remenik

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