Probability in Bahia Meeting

Universidade Federal da Bahia

Salvador, October 6-10, 2014

Talks

Bernardo Nunes Borges de Lima (UFMG)

Title: Embedding binary sequences into Bernoulli site percolation on $\mathbb{Z}^3$

Abstract: We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on \(\mathbb{Z}^d\) with parameter $p$. In 1995, I. Benjamini and H. Kesten proved that, for $d \geq 10$ and $p=1/2$, all sequences can be embedded, almost surely. They conjectured that the same should hold for $d \geq 3$. We consider $d \geq 3$ and $p \in (p_c(d), 1-p_c(d))$, where $p_c(d)<1/2$ is the critical threshold for site percolation on $\mathbb{Z}^d$. We show that there exists an integer $M = M (p)$, such that, a.s., every binary sequence, for which every run of consecutive {$0$s} or {$1$s} contains at least $M$ digits, can be embedded. Joint work with M. Hilário (UFMG), P. Nolin (ETH) and V. Sidoravicius (IMPA)



Robert Morris (IMPA)

Title: Solving extremal problems in random sets

Abstract: Many classical theorems in combinatorics take the following form: "All graphs / sets of integers avoiding a certain 'forbidden' sub-structure have size at most $x$." For example, Turán's theorem characterizes the maximum-size $H$-free graphs on $n$ vertices if $H$ is a clique, and Szemerédi's theorem states that every set of integers of positive density contains a $k$-term arithmetic progression. In recent years, there has been substantial interest in proving `sparse random analogues' of these classical theorems. For example, one of many conjectures made by Kohayakawa, Łuczak and Rödl in the mid-1990s states that Szemerédi's theorem continues to hold (with high probability) in a $p$-random subset of $\{1,\ldots,n\}$ if $p \gg n^{-1/(k-1)}$. Many of these conjectures were finally proven in 2009 by Schacht, and independently by Conlon and Gowers. In this talk I will recall some of these results, and discuss a third method of attacking such problems, which also allows one to prove other related `counting' results (How many $H$-free graphs are there?) and `structural' results (What does a typical $H$-free graph of a given density look like?). The basic unifying theme is a deterministic statement about the independent sets in $k$-uniform hypergraphs. Joint work with József Balogh and Wojciech Samotij.



Marielle Simon (ENS-Lyon)

Title: Macroscopic behavior for a disordered chain of harmonic oscillators, perturbed by a stochastic noise

Abstract: This talk will focus on scaling limits for interacting particle systems, evolving according to an energy-conservative dynamics. In a general context, one of the ultimate goals of statistical mechanics is to derive the macroscopic evolution of energy from a microscopic dynamics given by a chain of coupled oscillators. This is expected to hold through a diffusive space-time scaling limit. I will present two such microscopic models, both involving harmonic oscillators perturbed by a stochastic noise, one with equal constant masses, the other with random i.i.d. masses. I will investigate the macroscopic behavior of the system under two different aspects: first, the so-called hydrodynamic equations, and second, the macroscopic fluctuations of energy.



Augusto Teixeira (IMPA)

Title: Random Walk on Random Walks

Abstract: In this talk we will discuss a random walk evolving over a one-dimensional dynamic random environment, consisting of a collection of independent particles performing simple symmetric random walks, starting at a Poisson distribution with density $r > 0$. At each step, the random walk performs a nearest-neighbour jump, moving to the right with probability $p_v$ when it is on a vacant site and probability $p_o$ when it is on an occupied site. Under some assumptions on these parameters, we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviations bound. We will discuss why this model is interesting and what are the main challenges one has to overcome to analyze it. This talk will be based in a joint work with M. Hilário, F. den Hollander, V. Sidoravicius and R. Soares dos Santos



Daniel Ahlberg (IMPA)

Title: Scaling limits for the threshold window: When does a monotone Boolean functions flip its outcome?

Abstract: Let f be a monotone Boolean function on n variables and $\eta_p$ the canonical monotone coupling of an element in $\{0,1\}^n$ chosen according to product measure with intensity p. The random point at which $f(\eta_p)$ flips from being $0$ to $1$ as $p$ increases is often concentrated near a specific value thus exhibiting a threshold phenomena. For a sequence of such Boolean functions, we peer closely into this threshold window and consider the limiting distribution of this random point where the Boolean function switches from being $0$ to $1$. We determine this distribution for a number of the Boolean functions which are typically studied.



Fábio Júlio Valentim (UFES)

Title: The Relative Entropy Method and the Hydrodynamic limit for fast diffusion equation

Abstract: After a short introduction to basic ideas about to the subject, we will obtain the fast diffusion equation as hydrodynamic limit of a zero-range process with symmetric unit rate $g$. The fast diffusion effect comes from the fact that the diffusion coeficient $D(\rho)$ goes to infinity as $\rho\rightarrow{0}$. In order to capture this explosion we consider a model with a typically high number of particles per site. We follow the Relative Entropy method to prove the hydrodynamic limit.



Glauco Valle da Silva Coelho (UFRJ)

Title: Hydrodynamic limit for exclusion processes with slow bounds of constant rates in dimension $d\geq 2$

Abstract: We consider symmetric exclusion processes on the discrete $d$-dimensional discrete Torus associated to a closed hypersurface which slows down particles movement. We show that these nonhomogeneous systems have hydrodynamic limit under diffusive scaling whose hydrodynamic equations are of parabolic type associated to a sort of $d$-dimensional Krein-Feller operator. Joint work with Felipe Melo.



Wojtek Samotij (Tel Aviv University)

Title: The loop $O(n)$ model

Abstract: A loop configuration in the hexagonal (honeycomb) lattice is a finite subgraph of the lattice in which every vertex has degree 0 or 2, so that every connected component is isomorphic to a cycle (loop). The loop $O(n)$ model on the hexagonal lattice is a random loop configuration, where the probability of a given loop configuration is proportional to $x^{\sharp\, \textrm{edges}} n^{\sharp \,\textrm{loops}}$, where $x,n>0$ are parameters called the edge-weight and the loop-weight, respectively. We show that for sufficiently large $n$, the probability that a fixed hexagon is surrounded by a loop of length $k$ decays exponentially with $k$. In the same region of parameters, we also show a phase transition from a disordered phase to an ordered phase. No prior knowledge in statistical mechanics will be assumed and all notions will be explained. Joint work with Hugo Duminil-Copin, Ron Peled, and Yinon Spinka.



Patrícia Gonçalves (CMAT/PUC-Rio)

Title: A microscopic disorder in the SSEP: hydrodynamics and phase transitions

Abstract: In this talk we will introduce the SSEP with two types of disorders: either at a bond or at a site. We will consider the SSEP evolving on the one dimensional discrete torus $\mathbb{T}_n$ with $n$ sites. We attach a clock to each bond of $\mathbb{T}_n$, all the clocks being independent and exponential distributed with parameter $1$. After a ring of a clock, the particles at the bonds exchange positions. First, we perturb this dynamics by introducing a bond disorder. For that purpose, we only change the parameter of the clock corresponding to the jumps between $-1$ and $0$, and we take it equal to $\alpha/n^\beta$, $\alpha>0$ and $\beta\geq 0$. This means that microscopically, as beta increases the more difficult is the passage of particles across the bond $[-1,0].$ We will present the hydrodynamics for this model and several phase transitions appearing by changing the value of $\beta$. Second, we will discuss the more challenging case, the site disorder.



Manuel Stadlbauer (UFBA)

Title: Boundaries for random walks on groups with stationary increments

Abstract: Random walks on groups with independent increments are classical objects in probability theory which are also subject of recent research, like, e.g., local limits for random walks on hyperbolic groups by Gouezel or recent results on the realization of random walks with given Furstenberg entropy. In the talk, I will present a class of random walks with stationary increments which generalizes the classical object. By applying ideias from dynamical systems, that is by establishing a type of Perron-Frobenius theorem, it is possible to obtain an explicit construction of the family of eigenfunctions of the associated Markov operator. In most cases, this family of eigenfunctions is non-trivial, giving rise to a new construction of the boundary of the random walk.



Stefano Olla (Université Paris-Dauphine)

Title: Isothermal and adiabatic thermodynamics transformations from hydrodynamic limits.

Abstract: Isothermal and adiabatic transformations are the basic thermodynamics transformations composing Carnot cycles. I will illustrate how to obtain them from a microscopic dynamics under a diffusive space-time scaling limit. The deterministic irreversible transformations are given by a set of diffusive equations, and the quasi-static reversible transformations are obtained by a further time scaling for limiting slow variation of the applied force.



Paulo Ruffino (UNICAMP)

Title: Averaging principle on foliated space: application to the topology of submanifolds

Abstract: Consider an SDE on a foliated manifold such that each trajectory lays on a single leaf of the foliation. We investigate the effective behaviour of a perturbation of order $\epsilon$ in a direction transversal to the leaves, hence destroying the foliated structure of the trajectories. An average principle is shown to hold such that, rescaling the time, the vertical (transversal) coordinate converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to invariant measures on the leaves, as $\epsilon$ goes to zero. An estimate of the rate of convergence is given. These results generalize the geometrical scope of previous approaches, including completely integrable stochastic Hamiltonian system (X.M. Li, Nonlinearity 2008). We apply this result to Brownian motion on compact submanifolds of Euclidean spaces, such that the coefficients of the corresponding deterministic ODE's in the vertical coordinate are given by the Euler characteristics of the submanifolds.



Claudio Landim (IMPA)

Title: Macroscopic fluctuation theory

Abstract: Stationary non-equilibrium states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new phenomena. Within a statistical mechanics approach, these states have been the subject of several theoretical investigations, both analytic and numerical. The macroscopic fluctuation theory, based on a formula for the probability of joint space-time fluctuations of thermodynamic variables and currents, provides a unified macroscopic treatment of such states for driven diffusive systems. We give a detailed review of this theory including its main predictions and most relevant applications.



Freddy Hernandez (UFF)

Title: Non gradient method on spheres

Abstract: The purpose of this talk is to illustrate the so called non gradient method introduced by S.R.S Varadhan in the early nineties, through a concrete example, namely, an equilibrium fluctuation result. In particular, we will try to highlight some geometric aspects inherent to the method.



Gonzalo Fiz Pontiveros (The Hebrew University of Jerusalem)

Title: The triangle-free process

Abstract: In general, a random graph process consists of a randomly chosen sequence of edges $e\in E(K_n)^{\mathbb{N}}$, and a (random or deterministic) rule which determines the graph $G_m$ as a function of the sequence $(e_1,\ldots,e_m)$ (and often just of the graph $G_{m-1}$ and the edge $e_m$). The study of these objects has exploded in recent years, as the ubiquity of random-like networks in nature has come to the attention of the scientific community. Particularly well-studied processes include the ‘preferential attachment’ models of Barab\'asi and Albert, and the so- called ‘Achlioptas processes’ introduced by Achlioptas in 2000, and studied most notably by Riordan and Warnke. We mention also the very general class of ‘inhomogeneous random graphs’ introduced recently by Bollob ́ as, Janson and Riordan. Consider the following random graph process $(G_m)_{m\in \mathbb{N}}$ on vertex set $[n] = \{1,\ldots, n\}$. Let $G_0$ be the empty graph and, for each $m\in\mathbb{N}$, let $G_m$ be obtained from $G_{m-1}$ by adding a single edge, chosen uniformly from those non-edges of $G_{m-1}$ which do not create a triangle. The process ends when we reach a maximal triangle-free graph; we denote by $G_{n,\Delta}$ this (random) final graph. A technique which has proved extremely useful in the study of random graph processes is the so-called ‘differential equations method’, which was introduced by Wormald in the late 1990s. In this method, the idea is to ‘track’ a collection of graph parameters, by showing that (with high probability) they closely follow the solution of a corresponding family of differential equations. The aim of this talk is to give an overview of this method and to present recent results on the triangle-free process in joint work with R. Morris and S. Griffiths.



Posters

Diogo Bessam (PUC-Rio)

Title: Large deviations rates in a Gaussian setting: the role of the Cameron-Martin space.

Abstract: TWe discuss the use of the Cameron-Martin space of a Gaussian measure in the description of the large deviations rates in a Gaussian setting. We further illustrate the main ideas by looking at the Freidlin-Wentzell upper and lower estimates for $\varepsilon X$, as $\varepsilon \rightarrow 0$, where $X$ is a $R^d$ Gaussian vector and by constructing the Cameron-Martin space associated to $X$. Three classical examples of large deviations rates in this setting are surveyed and related to one another.



Caio Teodoro de Magalhães Alves (Unicamp)

Title: The soft local times method.

Abstract: We present a coupling method recently developed by Augusto Teixeira and Serguei Popov in ``Soft local times and decoupling of random interlacements’'. It consists in using a single realization of a Poisson point process in a suitably defined space to generate multiple probability measures we wish to compare. This method has been successfully used in the decoupling of processes that exhibit strong long range correlations, such as the cover time of the random walk on the discrete torus and the random interlacements model.



Pedro Franklin (UFMG)

Title: Processos de Crescimento com Obstáculos Paralisantes.

Abstract: Considere um sistema que possua substâncias verdes, brancas e vermelhas. As substâncias verdes são ativas, as vermelhas são paralisantes e as brancas são passivas. As substâncias verdes podem crescer e se juntar a outras substâncias verdes para formar os aglomerados verdes; as substâncias vermelhas são inativas enquanto isoladas, mas ao primeiro contato com algum aglomerado verde, a substância vermelha invade esse aglomerado paralisando seu crescimento tornando em seguida todas as suas substâncias verdes em vermelhas. As partículas brancas não crescem e em contato com alguma substância verde transformam-se também em substâncias verdes. O principal objetivo da exposisão será apresentar e estudar esse modelo. Vamos discutir se o modelo descrito acima está bem definido e se podemos falar alguma coisa da distribuição do tamanho dos aglomerados verdes. Será mostrado que se a densidade inicial das partículas brancas for pequena e a das partículas vermelhas for maior que zero, então o modelo estará bem definido e a distribuição do tamanho dos aglomerados verdes possui decaimento exponencial. Todo o conteúdo da exposição faz parte do artigo Random spatial growth with paralysing obstacles de J. van den Berg, Y. Peres, V. Sidoravicius e M. E. Vares.



Kenkichi Tsunoda (Tokyo University)

Title: Large deviation principle from a hydrodynamic limit for an asymptotically degenerate particle system.

Abstract: Goncalves et al. considered a certain class of particle systems which derive the porous medium equation as a consequence of a hydrodynamic limit. Since the porous medium equation is degenerate, the equation looses its parabolic character. From this fact, the analysis of the macroscopic equation and the microscopic particle system becomes much complicated. In this poster, I report on the large deviation principle from the hydrodynamic limit derived in their paper.



João Tiago Assunção Gomes (IMECC, UNICAMP)

Title: Aubry set for asymptotically sub-additive potentials

Abstract: Let $T : X \to X$ be a continuous function on a compact metric space $(X, d)$ and let $F := \{f_n : X \to \mathbb{R}\}_{n\geq 1}$ be an asymptotically sub-additive sequence of potentials. Define the ergodic maximizing value as $$\beta[F]\; :=\; \max\Big\{ \lim_{n\to\infty} \frac{1}{n}\int f_n d\mu\; :\; \mu \textrm{ is } T-\textrm{invariant probability}\Big\}$$ and denote by $\mathcal{M}_{\max}(F)$ the set of maximizing measures (that is, the $T$-invariant probabilities that attain the above maximum). We generalize for this context the notion of Aubry set, first introduced in ergodic optimization by [Contreras, Lopes, Thiellen]. Our main result provide a sufficient condition for the Aubry set, denoted by $\Omega(F)$, be a maximazing set: $$\nu\in \mathcal{M}_{\max}(F) \leftrightarrow \textrm{supp}\; \nu\subset \Omega(F).$$



Leandro Batista Morgado (Unicamp, Brazil)

Title: Decomposition of stochastic flows by Marcus equation and ``stop and go" technique

Abstract: The decomposition of a SDE flow in horizontal and vertical foliations can be substantially extended if we take as integrator an appropriate semimartingale with jumps, using the stochastic differential equation proposed by Marcus. The main idea is to define this integrator in a good way, such that the solution of Marcus equation jump over the critical points of the decomposition. Besides that, we propose the ``stop and go" stochastic differential equation. This equation is defined from another SDE, and its solution coincides with the original, unless in a subset S ⊂ R, where the process remains stationary. Then, we use the ``stop and go" technique to obtain a general result about the decomposition of a SDE flow.



Renata de Moura Issa Vianna (UFBA Brazil)

Title: KMP MODEL AND FOURIER’S LAW

Abstract: The KMP model is a classical interacting model consisting of a chain of one-dimensional uncoupled harmonic oscillators which interchange energy through an stochastic process. Each bond has a Poisson clock. Each time the bond rings, the two neighbors oscillators redistribute energy in a uniform way. Besides, the system is in contact with reservoirs at the boundaries. In this poster we present the study in progress of KMP model and open problems to be considered.



Support