# 2019

## 3/4/2019 - Jean-Marc Azais

Université de Toulouse

Repulsion or attraction  of critical points of random fields.

## 10/4/2019 - Alejandro Ramírez

Título: Paseos al azar en desorden bajo y alto

Resumen: Consideramos paseos al azar en desorden bajo en $\mathbb Z^d$. Para dimensiones $d\ge 4$,

mostramos que existe una transición de fase que depende de la intensidad del desorden y que

se expresa como una igualdad entre las funciones de tasa promediada y casi segura de los

principios de grandes desvíos correspondientes. En dimensión $d=2$ probamos que existe un escalamiento

universal del tiempo, espacio y desorden que converge a la ecuación del calor estocástica. Esta charla se

basa en trabajos conjuntos con  Bazaes, Mukherjee y  Saglietti, y con Moreno y Quastel.

## 20/2/2019 - Sergey Foss

Heriot-Watt University, Edinburgh, UK and Novosibirsk State University, Russia

Tail asymptotics for the supremum of a random walk.

Let S_n=X_1+...+X_n, n=1,2,... be the sums of i.i.d. random variables. Assume M = \sup_n S_n to be a.s. finite.

We are interested in the asymptotics for P(M>x) as x \to \infty and analyse 5 possible scenarios, including the 2 classic ones.

# 2018

## 12/12/2018 - Conrado Freitas Paulo Da Costa (Leiden University)

### On the strong law of large numbers for RWCRE under general ressampling maps

The Random Walks in Cooling Random Environments arose in the context of Perturbations of Random Walk in Random Environments. The cooling random environment is obtained by defining a sequence of ressampling times, the ressampling map, where we independently select a new environment (all sites have their transition kernels reassigned). This construction leads to a random walk that can be seen as a patchwork of independent pieces of different length of the static RWRE. The original motivation was to study this model when lengths between ressampling times diverges. In this talk we will examine the strong law of large numbers when this does not occur.

This is a Joint work (in progress) with Luca Avena.

## 05/12/2018 - Pablo Ferrari (UBA - CONICET)

### Soliton decomposition of excursions

I will describe recent work with Davide Gabrielli on the decomposition of excursions of a nearest neighbor random walk. We show that he soliton decomposition is equivalent to a branch decomposition of the Neveu-Aldous tree associated to the excursion. As a consequence, we show that the number of k-branches of the tree attached to each k-slot is geometrically distributed and that those numbers are independent.

## 28/11/2018 - Nahuel Soprano Loto (Universidad de Buenos Aires)

### Acerca de un modelo general de tipo Brunet-Derrida

La charla es acerca de un modelo de ramificación-selección de tipo Brunet-Derrida, introducido en [1], en el que tanto las tasas de ramificación como los mecanismos de selección dependen, en forma general, de los estadísticos de orden de las partículas involucradas. Dominando estocásticamente por procesos auxiliares, se exhibe explícitamente la velocidad asintótica del modelo. Para la cota inferior se define una familia de procesos que tienen a la ecuación de F-KPP por límite hidrodinámico.

Se trata de un trabajo en preparación en colaboración con Pablo Groisman.

[1] P. Groisman, M. Jonckheere, J. Martínez; Hydrodynamic limit and selection principle for a branching-selection particle system in the F-KPP class; en preparación.

## Fraser Daly

### Approximation by geometric sums: Markov chain passage times and queueing models

Sums of a geometrically distributed number of IID random variables occur in applications in many areas, including insurance, queueing theory and statistics. In this talk we will consider several settings in which random variables of interest may be successfully approximated by such geometric sums. In each case, we will give explicit error bounds (derived using Stein's method) to quantify the approximation results. Our main application is to passage times for stationary Markov chains, where we consider two different approximations. We will also demonstrate how our error bounds may be simplified under certain assumptions. This is illustrated by a brief look at approximation results for some performance measures of the M/G/1 queueing model.

## Seva Shneer

### Stability conditions for a discrete-time decentralised medium access algorithm

We consider a stochastic queueing system modelling the behaviour of a wireless network with nodes employing a discrete-time version of the standard decentralised medium access algorithm. The system is unsaturated -- each node receives an exogenous flow of packets at the rate λ packets per time slot. Each packet takes one slot to transmit, but neighbouring nodes cannot transmit simultaneously. The algorithm we study is standard in that a node with empty queue does not compete for medium access and the access procedure by a node does not depend on its queue length, as long as it is non-zero. Two system topologies are considered, with nodes arranged in a circle and in a line. We prove that, for either topology, the system is stochastically stable under condition λ<2/5. This result is intuitive for the circle topology as the throughput each node receives in a saturated system (with infinite queues) is equal to the so-called parking constant, which is larger than 2/5. (The latter fact, however, does not help to prove our result.) The result is not intuitive at all for the line topology as in a saturated system some nodes receive a throughput lower than 2/5.

This is a joint work with Sasha Stolyar (UIUC).

## 14/11/2018 - Gérald Tenenbaum (Institut Elie Cartan - Université de Lorraine - Francia)

### On arithmetical processes

The talk will be mainly devoted to survey arithmetical models of some random processes. In other words, we shall describe situations in which arithmetic functions, depending on the multiplicative structure of integers and considered as random variables on the set of the first N integers, can be used to quantitatively approximate classical and less classical random processes from probability theory.

## 17/10/2018 - Pablo Ferrari (UBA-Conicet)

### Boson point process, Gaussian loop soup and interlacements

We prove that the infinite volume boson point process at any density is the point marginal of a spatial random permutation (X,p), where X is a locally finite random subset of Rd and p : X -> X is a bijection. The law of (X,p) is Gibbs with respect to the Hamiltonian H(X,p) = \alpha sum_{x\in X} ||x-p(x)||^2; \alpha is the temperature.

There is a density rho_c such that for point density \rho\le\rho_c the spatial random permutation is a Poisson process of loops of a random walk with Gaussian increments, called Gaussian loop soup. This representation immediately implies that the marginal law of X is a permanental process. \rho_c is infinite for d=1,2 and rho_c is finite for d\ge 3.

For d\ge 3 and \rho>\rho_c the spatial random permutation is a Poisson process of infinite trajectories of a random walk with Gaussian increments at point density \rho-\rho_c superposed to an independent Gaussian loop soop at density \rho_c. Using the representation we easily compute the multi point density of X.

We also present an apparently novel construction of the random interlacements.

This is joint work with Inés Armendáriz and Sergio Yuhjtman.

## 26/09/2018 - Marcelo Costa (Universidad de Buenos Aires)

### Cooperative models of stochastic growth

Se puede ver el resumen de la charla acá.

## 29/08/2018 - Emanuel Ferreyra (Universidad de Buenos Aires)

### Dinámica SIR con vacunación óptima en un grafo aleatorio

Trabajo conjunto con Matthieu Jonckheere y Juan Pablo Pinasco.

## 22/08/2018 - Conrado Freitas Paulo Da Costa (Leiden University)

### Limit behaviors of the Random walk in the Cooling Random Environment (RWCRE)

The goal of the talk is to explain the Strong Law of Large Numbers (SLLN) and the Large deviation Principle (LDP) for the RWCRE. We start by introducing the model, and recalling a few traits of the classical Random Walk in Random Environment (RWRE). Next we discuss the basic ideas used in the proof of the SLLN and  LDP for the RWCRE.

References: https://arxiv.org/abs/1610.00641 https://arxiv.org/abs/1803.03295

## 15/08/2018 - Lou Kondic (New Jersey Institute of Technology)

### Interaction networks in particulate-based systems: persistence, percolation, and universality

Interaction networks are mesoscale structures that form spontaneously as particulate-based systems (such as granulars, emulsions, colloids, foams…) are exposed to shear, compression, or impact.  The presentation will focus on few different but closely related questions involving properties of these networks: (i) Are the networks universal, with their properties independent of those of the underlying  particles?  (ii)  What are percolation properties of these networks, and can we use the tools of percolation theory to explain their features? (iii) How to use algebraic topological tools, and in particular persistence homology based approach to quantify the static and dynamics properties of these networks? The presentation will focus on the results of molecular dynamics simulations to discuss these questions and (currently known) answers, but we will also discuss how to relate and apply these results to physical experiments.

## 08/08/2018 - Leonardo Rolla (Universidad de Buenos Aires, New York University Shanghai)

### Recurrence and transience for the frog model on trees

We will discuss the transition from transience to recurrence in the frog model on trees, and how it depends on the initial distribution. Joint with Tobias Johnson.

## 01/08/2018 - Boguslaw Zegarlinski (Imperial College London)

### Some new results on hypoelliptic diffusion

My talk will be about controlling smoothing and ergodicity for some models of hypoelliptic diffusion. This will include recent results on hypercontractivity of some symmetric diffusion as well as diffusions defined by parabolic PDE with coefficients dependent by a meanfield.

## 11/07/2018 - Roberto Fernández (New York University Shanghai)

### Escape de trampas metaestables y la noción de estado metaestable

El comportamiento metaestable está asociado a "trampas" que atrapan al sistema durante largo tiempo, y de las que se emerge en un tiempo aleatorio típicamente distribuído con una ley exponencial. La charla presentará el alcance y las limitaciones de los diferentes métodos de estudio matemático de estos fenómenos, y discutirá en detalle el enfoque denominado "pathwise approach" que es quizás el más fiel a la intuición física. Se expondrá una teoría relativamente reciente que se aplica a fenómenos generales de metaestabilidad, prescindiendo de la hipótesis habitual de reversibilidad, y haciendo referencia a genuinos estados y no sólo a configuraciones metaestables.

## 04/07/2018 - Sergio Yuhjtman (Universidad de Buenos Aires)

### Permutaciones aleatorias asociadas al gas de Bose libre

Es posible estudiar la mecánica estadística de los sistemas de bosones cuánticos a través de modelos de permutaciones  aleatorias. El caso más sencillo es el gas de Bose libre. András Sütö demostró (entre 1993 y 2001) que la condensación de Bose-Einstein coincide con la aparición de ciclos infinitos en las permutaciones aleatorias asociadas. Junto con Inés Armandariz y Pablo Ferrari, logramos construir las medidas de probabilidad a volumen infinito (en R^d) correspondientes a estas permutaciones aleatorias. Los ciclos infinitos pertenecen a una familia de procesos conocida como "random interlacements".

## 27/06/2018 - Daniel Fraiman (Universidad de San Andrés)

### Criticalidad auto-organizada: el modelo de evolución de Bak-Sneppen y un modelo de compra-venta.

[1] P. Bak, K. Sneppen. Punctuated equilibrium and Criticality in a Simple Model of Evolution. Phys. Rev. Lett. 71, 4083 (1993).

[2] H. Luckock. A steady-state model of the continuous double auction. Quantitative Finance, 3, 385-404 (2003).

[3] D. Fraiman. Bak-Sneppen model: Local equilibrium and critical value. Phys. Rev. E, 97(4), 042123 (2018).

[4] D. Fraiman. Self-organized criticality auction model for selling products in real time. (arXiv:1805.09763).

## 13/06/2018 - Facundo Sapienza (UBA - Aristas)

### Geodesics in First Passage Percolation and Distance Learning

Given a discrete set X of points in the Euclidean space, ​how can we define a distance between elements of X that takes into account the underlying structure of X? We will introduce the "Fermat's distance" and its estimator, a new metric with applications in Machine Learning and Statistics [1].

​When X is given by a homogeneous Poisson point process, this problem ​has been studied in [2,3], where the authors prove the convergence of the estimator of Fermat's distance in the context of Euclidean First Passage Percolation. We will show how these results can be generalized to the case when X is an i.i.d sample with density distribution supported on a lower dimensional manifold, which is the typical scenario in real data. How does the geodesics of this new process like? What is the macroscopic limit of the estimator? How can we make these results meaningful in real applications?  We will also see how this distance is related to Fermat's Principle in optics that states which is the path followed by light in a non-homogeneous media.

[1] F. Sapienza, P. Groisman, M. Jonckheere; “Weighted Geodesic Distance Following Fermat Principle”; 6th International Conference on Learning Representations.

[2] C.D. Howard and C.M. Newman;  “Euclidean models of first-passage percolation”; Probability Theory and Related Fields, 108(2):153–170, 1997.

[3] C.D. Howard and C.M. Newman; “Geodesics and spanning trees for Euclidean first-passage percolation”; Annals of Probability, vol 29, no. 2, pp. 577-623.

## Elene Anton (Institut de Recherche en Informatique de Toulouse)

### Analysis of the impact of mobility in cellular networks

In order to analyse the impact of mobility in wireless networks, we look to a K parallel server

system with mobility, which means that when a user arrives at the system, it fulfils an independent

random walk among the server until it gets all the service it requires. We take Markovian

assumptions. Our main goal was to analyse the impact of mobility in this network.

For this purpose, we introduce parameter α ≥ 0, the mobility speed. We analyse the system

where users move among the servers at rates αr_{ij} . We have first defined the system when mobility

speed is infinite. Then, we have proved that when α → ∞ the stationary distribution of the limit

of the main system exists and is unique. In this context, we have characterized this limit. Furthermore,

we have seen that in the limit the system decomposes into two independent components: on

the one hand the total number of users in the system (M/M/1) and on the other, conditionally on

the total number of users present in the system, the distribution of these users among the serves

(the multinomial distribution). Finally, we have proved, against what it is written in literature,

that the performance of the system strongly depends on mobility speed.

## Olivier Carton (Université Paris Diderot)

### Normal numbers with constraints

We first recall the definition of normality which is a kind of (very) weak randomness.  We consider normal number digit dependencies in their expansion in a given integer base.  We quantify precisely how much digit dependence can be allowed such that, still, almost all real numbers are normal.  In some cases, we are even able to prove that still, almost all real numbers are absolutely normal.

## 30/05/2018 - Federico Holik (Insituto de Física La Plata - UNLP - CONICET)

En esta charla se discutirán las probabilidades cuánticas desde el punto de vista de una generalización no conmutativa de la teoría de la medida. El foco estará en distintos aspectos geométricos. Se describirá cómo extender este abordaje a otras teorías probabilísticas, y discutir sus relaciones con la teoría de la información cuántica.

## 16/05/2018 - Leonardo Rolla, Universidad de Buenos Aires / NYU Shanghai

H-Transform of Simple Random Walk on Z²

The Simple Symmetric Random Walk on Z² is recurrent, but barely. Its h-transform is a homogeneous Markov chain which corresponds to the walk being in some sense conditioned on never returning to the origin. It is a natural object that appears in many different situations. In this talk we consider some of the properties of this process. Based on ongoing work with S. Popov.

## 02/05/2018 - Monia Capanna, Universidad de Buenos Aires

Critical fluctuations in the SIR model

In this talk I will analyze a model for the spread of an epidemics among a finite population containing susceptible, infected and removed individuals (SIR). The microscopic scenario consists of an interacting particle system in the discrete torus in which the possible states for the sites are 0=susceptible, 1=infected, and -1=recovered. Each infected particle infects a susceptible particle with a rate that depends on their relative distance in such a way that nearby sites are infected with a greater intensity than those further away. Infected particles recover with constant rate equal to 1.

I will prove that the microscopic densities of susceptible and infected individuals converge in the hydrodynamic limit to the solution of two coupled PDE's. I will find an implicit expression for the final survivor density of the limit equation and I will analyze the fluctuations from the hydrodynamic limit of the system showing a critical behaviour for some values of the parameters.

## 21/03/2018 - Eric Cator, Radboud University

### Ergodic theory of stochastic Burgers equation in non-compact setting

In this talk I will explain recent results about the existence of a one-force-one-solution principle for the stochastic Burgers equation in a non-compact (but homogeneous) setting. In recent years several results were proved for stochastically forced Burgers equation in (essentially) compact settings, showing that there exists global solutions that act as attractors for large classes of initial conditions. However, extending these results to truly non-compact settings was not possible using the same methods, and it was even conjectured by Sinai that the results would not hold in that case. Using results from First and Last Passage Percolation, first developed by Newman et al., we were able to prove the one-force-one-solution principle for a Poisson forcing on the real line. This is joint work with Yuri Bakhtin and Konstantin Khanin.