17/7/2019 - Herbert Spohn
Technical University Munich
Hydrodynamics of the classical Toda chain
Over the past four years there have been many activities, mostly on the quantum side, to study the dynamics of integrable many-particle systems at non-zero temperatures. Apparently they share common features. The classical Toda chain will serve as a road map in illustrating the recent advances.
11/7/2019 - Avelio Sepúlveda
Université Lyon 1
Level sets of the two-dimensional GFF
In this talk, we will give an overview of the theory of level sets of the two-dimensional continuum Gaussian free field (GFF). This theory studies points where, in a certain sense, the GFF takes finite values. We will present the geometric and percolative properties of these sets, and we will relate them to the critical loop soup and to Poisson point process of boundary-to-boundary excursions. Based on joint works with Juhan Aru, Titus Lupu and Wendelin Werner.
10/7/2019 - Arvind Ayyer
Indian institute of science
Correlations in the multispecies exclusion process
Recently, multispecies exclusion process on finite one-dimensional
lattices turned up unexpectedly in the study of a natural transient
random walk on an affine Weyl group by Thomas Lam. For the case of Type
A, Lam conjectured an explicit formula for the limiting direction of the
walk. In joint work with Svante Linusson (TAMS 2017), we proved this
conjecture using the stationary distribution formulas of P. Ferrari and
J. Martin. More recently, in joint work with Aas, Linusson and Potka, we
have determined limiting directions for Lam's walks on types B, C and D
by understanding the stationary distribution of a new exclusion process
we call the D*-TASEP. I will give an overview of these results and
explain the key ideas involved in the proofs.
4/7/2019 - Jorge Kurchan
École normale supérieure
Abstract
Los modelos estocasticos de particulas que difunden en una red uni-dimensional
tienen un operador evolucion (Perron-Frobenius) que puede escribirse como una cadena de spins
cuanticos. Por ejemplo, el SSEP (simple symmetric exclusion process) esta representado por una
cadena de spins, del grupo SU(2).
Hace un tiempo mostramos que otro modelo conocido, debido a Kipnis Marchioro y Presutti, donde
lo que se transporta energia, corresponde a una cadena de spins SU(1,1).
Asi presentados, estos modelos aparecen en fisica del solido y de altas energias. Es interesante ver
como se conectan estos dos campos con el de la probabilidad.
Duality and Hidden Symmetries in Interacting Particle Systems
Non-compact quantum spin chains as integrable stochastic particle processes
25/6/2019 - Roberto Fernández
New York University, Shanghai
joint work with Nahuel Soprano-Loto (UBA)
Uniqueness and non-uniqueness criteria based on the generalized FK representation
We use the Edwards-Sokal generalization of the FK random-cluster representation to establish criteria for both absence and existence of phase transitions. The uniqueness criterion applies to general spin models with summable interactions and it extends a criterion due to Alexander and Chayes, related to disagreement percolation. The criterion for existence of phase transitions, in turns, requires spins with an Abelian group structure and an interaction invariant under group products and displaying an energy gap. Furthermore, models must allow the definition of some generalized reflections connecting ground states. The criterion can be successfully applied to generalized clock and Ashkin-Teller models, and to gauge disorder effects in diluted models.
19/6/2019 - Santiago Saglietti
Technion, Israel
Weak convergence for the scaled cover time of the rooted binary tree
Abstract: We consider a continuous time random walk on the rooted binary tree of depth n with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by 2^{n+1}n and then centered by (log2)n-log n, the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a randomly shifted Gumbel random variable with rate one, where the shift is given by the sum of the limits of the derivative martingales associated with two negatively correlated discrete Gaussian free fields on the infinite version of the tree. The existence of the limit and its overall form were conjectured in the literature. Our approach is quite different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian free field. Joint work with Aser Cortines and Oren Louidor.
UBA
Caracterización del número de independencia asintótico de sucesiones de grafos aleatorios con grados dados
Resumen: Calcular el número de independencia (tamaño de los conjuntos de
independencia máximos de un grafo) es un problema NP-Difícil. En esta
charla mostraremos una condición suficiente para que un algoritmo local simple
construya asintóticamente conjuntos independientes máximos en sucesiones
de grafos aleatorios con grados dados. En los casos en los que se cumpla esta
condición, daremos también una caracterización del número de independencia
asintótico de estos grafos. Como una aplicación de los resultados discutidos,
se calculará el número de independencia asintótico de grafos Erdös-Rényi de
grado medio menor a e.
Heriot-Watt University
The critical greedy server on the integers
Abstract:
Each site of the one-dimensional integer lattice hosts a queue with arrival rate \lambda.
A single server, starting at the origin, serves its current queue at rate \mu until that queue
is empty, and then moves to the longest neighbouring queue. In the critical case
\lambda=\mu, we show that the server returns to every site infinitely often.
We also give an iterated logarithm result for the server's position.
In the talk I will try to explain the main ingredients in the analysis:
(i) the times between successive queues being emptied exhibit doubly exponential growth,
(ii) the probability that the server changes its direction is asymptotically equal to 1/4,
(iii) a martingale construction that facilitates the proofs.
This is joint work with Andrew Wade (Durham).
The contact process: recent results on finite-volume phase transitions
Abstract
The contact process is a model for the spread of an infection in a population. At any given instant of time, vertices of a graph (interpreted as individuals) can be either healthy or infected; infected individuals recover with rate 1 and transmit the infection to neighbours with rate lambda. On finite graphs, the infection eventually disappears with probability one. In many cases, the time it takes for this to occur depends sensitively on the value of the parameter lambda, and this finite-volume phase transition can be linked to a phase transition of the contact process on a related infinite graph. We will survey recent works in this direction, including some general results which hold on large classes of graphs, as well as results on specific graphs, such as finite d-ary trees and the random graph known as the configuration model.
This talk includes joint work with M. Cranston, T. Mountford, J.C. Mourrat, Bruno Schapira and Q. Yao.
UBA
Una introducción al método de Inferencia Variacional
Resumen:
Los métodos bayesianos son sumamente utilizados dentro del Aprendizaje
Automático. Algunos ejemplos de aplicación en problemas no supervisados
pueden encontrarse en el clustering de textos o la detección de
comunidades. En modelos muy complejos (es decir con muchas variables
latentes no independientes) puede resultar imposible calcular la
distribución a posteriori de las variables latentes.
El método de Inferencia Variacional propone aproximar la distribución a
posteriori de estos modelos por una distribución más simple, mediante la
optimización de un problema variacional en el espacio de distribuciones.
Este método resulta una poderosa alternativa al método Monte Carlo.
En esta charla recorreremos los fundamentos de este método, mencionando
también algunas aplicaciones del mismo.
Pontificia Universidad Católica de Chile
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.
Université de Toulouse
Repulsion or attraction of critical points of random fields.
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.
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.
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.
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.
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.
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).
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.
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.
Se puede ver el resumen de la charla acá.
En los últimos años, el problema del consenso de opiniones en una sociedad ha sido explorado por medio de modelos de partículas o agentes interactuantes. Por simplicidad, la mayoría de estos modelos asume que los contactos entre los individuos son homogéneos, aunque en la vida real existe una gran variedad de posibilidades que dependen del par interactuante. En este trabajo estudiamos un modelo de formación de opiniones en el que cada individuo en una población puede adoptar una de dos opciones posibles, estar a favor o en contra de un determinado tema. Para contemplar la heterogeneidad en las interacciones, le agregamos a cada individuo un factor de convicción en su opinión, de forma que la distribución de las convicciones y las opiniones evolucionan en forma acoplada. A diferencia de otros modelos de opiniones que incorporan agentes heterogéneos, en esta dinámica la heterogeneidad en la convicción emerge naturalmente como consecuencia de las interacciones entre pares de agentes. Encontramos que esta heterogeneidad produce una variedad de comportamientos colectivos en el sistema, como evoluciones rápidas al consenso y oscilaciones amortiguadas en los porcentajes de opiniones.
Trabajo conjunto con Matthieu Jonckheere y Juan Pablo Pinasco.
El objetivo principal es analizar cómo interactúa el proceso de vacunación con la propagación de una enfermedad, a partir de los estudios realizados en modelos de este tipo pero sin vacunas. Para esto, trabajaremos con una población infinita cuyas interacciones están dadas por un grafo estocástico de tipo Configuration Model. En cualquier momento de un período de tiempo continuo con horizonte finito, los individuos pueden estar en cualquiera de los cuatro estados: Susceptible, Infectado, Recuperado o Vacunado. Siguiendo un proceso de Poisson, los agentes interactúan con sus vecinos en el grafo, transmitiendo la enfermedad de los infectados a los susceptibles. Además, habrá una estrategia de vacunación que consiste en la tasa a la que un jugador se vacuna y que dependerá del grado del nodo. Nos enfocamos en cuatro medidas de la red de conectividad y probamos la convergencia de las medidas empíricas a un límite fluido determinado por un sistema contable de ecuaciones diferenciales que muestra la dependencia de la distribución inicial de grados en la propagación de la enfermedad. Una vez conseguido un sistema cerrado, analizamos la existencia de estrategia óptima tanto individual como social y la caracterizamos.
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
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.
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.
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.
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.
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".
En 1993 Per Bak y Kim Sneppen introdujeron un modelo dinámico para describir cómo evolucionan las especies[1]. El modelo es extremadamente simple y atrajo mucho interés científico. Se trata de un modelo de N especies interactuantes donde cada una está caracterizada por un valor de adaptación al medio, llamado fitness. El modelo supone una interacción a primeros vecinos en un arreglo unidimensional en forma de anillo con una condición inicial Uniforme(0,1). La dinámica de evolución es la siguiente, en cada paso temporal la especie menos adaptada (menor fitness) muere y también sus especies vecinas, y son reemplazadas por tres nuevas especies con fitness Uniformes(0,1). Luego de un estado transitorio el sistema arriba a un curioso equilibrio donde todas las especies tienen un valor de adaptación por arriba de un umbral pc. Ninguna especie sobrevive si su valor de adaptación es menor a este valor. Otro hecho curioso del modelo es que aparecen fenómenos de cascada (o avalancha) de extinción de especies. El modelo pertenece a la clase de modelos de criticalidad auto-organizada (SOC). Estos modelos resultan útiles para describir un conjunto diverso de fenómenos que incluyen la compra-venta de acciones o productos [2]. En esta charla discutiremos el modelo Bak-Sneppen [3], versiones simplificadas del mismo y un modelo de compra-venta [4].
[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).
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.
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.
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.
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.
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.
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.
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.