## Schedule

Probabilistic models for the (sub)Tree(s) of LifeAmaury Lambert, UPMC, France

The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model gene trees or species trees. We will start with stochastic models of tree shapes (finite trees without edge lengths), culminating in the β family of Aldous' branching models. We will next introduce real trees (trees as metric spaces) and show how to study them through their contour, provided they are properly measured and ordered. We will then focus on the reduced tree, or coalescent tree, which is the tree spanned by alive individuals/species at the same fixed time. We show how reduced trees, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. Beautiful examples of random combs include the Kingman coalescent and coalescent point processes. We will end up displaying some recent biological applications of coalescent point processes to the inference of species diversification.

OK
Extrema of logarithmically correlated Gaussian fields and related processesOfer Zeitouni, Weizmann Institute of Science, Israel

Ofer Zeitouni will discuss techniques that lead to the proof of convergence in distribution of the centered maximum of two dimensional Gaussian free field, the extremal field of the GFF, as well as related processes. A key element in the proof is an appropriate variation of the second moment method.

OK

## Plenary Talks

Percolation and local isoperimetric inequalities
Augusto Teixeira, IMPA, Brasil

Condensing zero range processes
Claudio Landim, IMPA, brazil

Percolation on the stationary distribution of the voter model on $Z^d$
Daniel Valesin, University of Groningen, The Netherlands

Branching in log-correlated random fields
David Belius, New York University, United States

Ergodic theory of stochastic Burgers equation in non-compact setting
Eric Cator, Radboud University Nijmegen, the Netherlands

Travelling front in a branching-selection process
Francis Comets, Paris-Diderot, France

The Bak-Sneppen Model of Biological Evolution and Related Models
Iddo Ben Ari, University of Connecticut, USA

Yaglom limits via Holley inequality
Pablo Ferrari, Universidad de Buenos Aires, Argentina

Traveling fronts and quasi-stationary distributions: microscopic behavior, simulation and selection principles.
Pablo Groisman, Universidad de Buenos Aires, Argentina

Optimal estimators of the mean: subgaussian tails from heavy tailed data
Roberto Imbuzeiro, IMPA, Brasil

Entropy and hypo-coercive methods in hydrodynamic limits
Stefano Olla, CEREMADE-Université Paris Dauphine, France

Percolation and local isoperimetric inequalitiesAugusto Teixeira, IMPA, Brasil

In this talk we will discuss some relations between percolation on a given graph $G$ and its geometry. There are several interesting questions relating various properties of $G$, such as growth or dimension, and the process of percolation on $G$. In particular one could look for conditions under which its critical percolation threshold $p_c(G)$ is non-trivial, that is: $p_c(G)$ is strictly between zero and one. In a very influential paper on this subject, Benjamini and Schramm asked whether it was true that for every graph satisfying $dim(G) > 1$, one has $p_c(G) < 1$. We will explain this question in detail, explaining what they meant by the dimension of a graph and we will present some results that have been recently obtained in this direction.

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Condensing zero range processesClaudio Landim, IMPA, brazil

We examine the nucleation phase of condensing zero range processes and their metastable behavior.

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Percolation on the stationary distribution of the voter model on $Z^d$Daniel Valesin, University of Groningen, The Netherlands

The voter model on $\Z^d$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When the model is considered in dimension 3 or higher, its set of (extremal) stationary distributions is equal to a family of measures $\mu_\alpha$, for $\alpha$ between 0 and 1. A configuration sampled from $\mu_\alpha$ is a field of 0's and 1's on $\Z^d$ in which the density of 1's is $\alpha$. We consider such a configuration from the point of view of site percolation on $\Z^d$. We prove that in dimensions 5 and higher, the probability of existence of an infinite percolation cluster exhibits a phase transition in $\alpha$. If the voter model is allowed to have long range, we prove the same result for dimensions 3 and higher. Joint work with Balazs Rath.

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Branching in log-correlated random fieldsDavid Belius, New York University, United States

I will discuss how log-correlated random fields show up in diverse settings, including the study of cover times, analytical number theory and random matrix theory. One way to understand this is that all these models possess an approximate branching structure. I will talk about how this insight can be used to prove certain results about extrema in the aforementioned settings.

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Ergodic theory of stochastic Burgers equation in non-compact settingEric Cator, Radboud University Nijmegen, the Netherlands

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.

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Travelling front in a branching-selection processFrancis Comets, Paris-Diderot, France

We consider a particle system introduced by E. Brunet and B. Derrida, which evolves according to a branching mechanism with selection of the fittest keeping the population size fixedd and equal to N. The particles remain grouped and move like a traveling front driven by a random noise with a deterministic speed. The model has an exact solution when the displacements are Gumbel distributed. Because of its mean-field structure, the model can be further analyzed for general laws as N increases. If the noise lies in the max-domain of attraction of the Weibull extreme value distribution the finite-size correction to the speed has universal features. Based on a joint work with Aser Cortines.

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The Bak-Sneppen Model of Biological Evolution and Related ModelsIddo Ben Ari, University of Connecticut, USA

The Bak-Sneppen model is a Markovian model for biological evolution was introduced as an example for Self-Organized Criticality. In this model, a population of size N evolves according to the following rule. The population is arranged on a circle, or more generally a connected graph. Each individual is assigned a random fitness, uniform on [0,1], independent of the other fitnesses. At each unit of time, the least fit individual, along with its neighbors are removed from the population and are then replaced by new individuals. Despite being extremely simple, the model is known to be very challenging, and the evidence for Self-Organized Criticality provided by Bak and Sneppen was obtained through numerical simulations. I will review the main rigorous results on this model (mostly due to R.Meester and his coauthors), present some new results and open problems. I will then turn to a recent and more tractable variant of the model, in which the spatial structure is relaxed, while the population size modeled by a random process. I will focus on the functional central limit theorem for the model, which has a somewhat unusual form.

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Yaglom limits via Holley inequalityPablo Ferrari, Universidad de Buenos Aires, Argentina

An absorbed Markov chain is conditioned not to be absorbed until time t. If the conditioned distribution at time t is the same as the initial distribution, then it is called quasi stationary distribution. Starting the chain with a fixed state, the Yaglom limit is the limit as t goes to infinity of the distribution of the chain at time t conditioned to non absorption. We show how to establish conditions for the domination of the trajectories which implies convergence of the Yaglom limit to the minimal quasi stationary distribution. The tool is Holley inequality, a result that uses local domination to prove global domination of measures.

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Traveling fronts and quasi-stationary distributions: microscopic behavior, simulation and selection principles.Pablo Groisman, Universidad de Buenos Aires, Argentina

We will review some common facts and differences between the traveling fronts phenomenon and that of quasi-stationary behavior in one dimension. In particular we will prove (for Lévy processes) the equivalence of the existence problem. We will also deal with the microscopic behavior. On the one hand Brunet-Derrida particle systems and on the other hand Fleming-Viot like systems. This has to do with the problem of simulation.

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Optimal estimators of the mean: subgaussian tails from heavy tailed dataRoberto Imbuzeiro, IMPA, Brasil

We discuss the problem of estimating the mean of a probability distribution over R from n i.i.d. samples. Our goal is to obtain estimators with uniform finite sample guarantees over large classes of distributions with finite variance, which nevertheless may be "fat tailed". Moreover, we want these estimators to have gaussian-like tails all the way down to extremely rare events (say with probability exponentially small in n). In this talk give conditions for such estimators to exist. Variants of our results were previously known for the case where estimating requires choosing a confidence level in advance, which is not desirable in many applications. Our estimators are shown to be optimal (up to constants) for large classes of distributions P, including all P with a given variance, all P that are symmetric around the mean, and all P satisfying higher-moment conditions. In each case, we also find the optimal range for gaussian-like tails, sometimes with sharp constants. Finally, we show that, if only second moments are assumed, then some degree of knowledge about the variance is necessary and sufficient to build good estimators. Joint work with Luc Devroye, Matthieu Lerasle and Gabor Lugosi.

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Entropy and hypo-coercive methods in hydrodynamic limitsStefano Olla, CEREMADE-Université Paris Dauphine, France

Relative Entropy and entropy production have been main tools in obtaining hydrodynamic limits Entropic hypo-coercivity can be used extend this method to dynamics with highly degenerate noise. I will apply it to a chain of anharmonic oscillators immersed in a temperature gradient. Stationary states of these dynamics are of ’non equilibrium’, and their entropy production does not allow the application of previous techniques. These dynamics model microscopically an isothermal thermodynamic transformation between non-equilibrium stationary states. Ref: http://arxiv.org/abs/1505.05002

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## Talks (30min)

Topological and statistical-mechanics properties of hierarchical systems
Elena Agliari, Sapienza, University of Rome, Italy

A percolation model without a phase transition
Erik Broman, Uppsala University, Sweden

Branching Random Walks and Multi-Type Contact-Processes on the Percolation Cluster of $Z^d$
Fabio Zucca, Politecnico di Milano, Italy

Zero Range Process with superlinear rates
Ines Armendariz, Universidad de Buenos Aires, Argentina

The distribution of the quasispecies
Joseba Dalmau, Ecole Normale Superieure, France

Variable speed Branching Brownian motion
Lisa Hartung, IAM, University of Bonn, Germany

Two dymensional layered systems at the mean field critical temperature
Maria Eulalia Vares, IM-UFRJ, Brazil

Explosion, implosion, and moments of passage times for continuous-time Markov chains : a semi-martingale approach
Mikhail Menshikov, University of Durham , Department of Mathematical Sciences, United Kingdom

Hydrodynamic Limit, Propagation of Chaos and Large Deviation for a continuous interacting spin system
Patrick Erich Mueller, IAM, University of Bonn, Germany

Is there a mutation threshold?
Rinaldo Schinazi, University of Colorado, USA

Topological and statistical-mechanics properties of hierarchical systemsElena Agliari, Sapienza, University of Rome, Italy

We consider statistical-mechanics models for spin systems built on hierarchical structures, which provide a simple example of non-mean-field framework. We show that the coupling decay with spin distance can give rise to peculiar features and phase diagrams much richer than their mean-field counterpart. In particular, we consider the Dyson model, mimicking ferromagnetism in lattices, and we prove the existence of a number of metastabilities, beyond the ordered state, which become stable in the thermodynamic limit. Finally, we show that the emergence of such meta-stabilities can be recast in purely stochastic terms, namely as the emergence of ergodicity breakdown for the Markov process generated by the adjacency matrix associated to the hierachical structure underlying the spin system.

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A percolation model without a phase transitionErik Broman, Uppsala University, Sweden

I will present some recent research (joint with J. Tykesson) on the so-called Poisson cylinder model. This is a percolation model which is non-local in the sense that the percolative objects are unbounded. We determine the exact connectivity properties for this model in Euclidean space as a function of the intensity u of the underlying Poisson process. As a consequence, some natural questions occur. For instance, what happens if we change the geometry of the underlying space? Thus, as a second step we investigated the connectivity properties of the model in hyperbolic space.

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Branching Random Walks and Multi-Type Contact-Processes on the Percolation Cluster of $Z^d$Fabio Zucca, Politecnico di Milano, Italy

In this talk we show that under the assumption of quasi-transitivity, if a branching random walk on $Z^d$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster $C_\infty$ of a supercritical Bernoulli percolation. When no more than k individuals per site are allowed, we obtain the k-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already k individuals are present. We show that local survival of the branching random walk on $Z^d$ also implies that for k sufficiently large the associated k-type contact process survives on $C_\infty$. This implies that the strong critical parameters of the branching random walk on $Z^d$ and on $C_\infty$ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated k-type contact processes. These results are extended to a family of restrained branching random walks, that is branching random walks where the success of the reproduction trials decreases with the size of the population in the target site. This is a joint work with D.Bertacchi.

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Zero Range Process with superlinear ratesInes Armendariz, Universidad de Buenos Aires, Argentina

We prove the existence of ZRP dynamics with different initial conditions and in general dimensions. We also prove, in 1d, that the limit of any ergodic translation invariant measure under the dynamics, as time tends to infinity, is the extremal invariant measure with the same density of particles. This is joint work with Enrique Andjel and Milton Jara.

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The distribution of the quasispeciesJoseba Dalmau, Ecole Normale Superieure, France

In 1971, Eigen proposed a deterministic model in order to model the evolution of an infinite population of macromolecules subject to mutation and selection forces. As a consequence of the study of Eigen’s model, two important phenomena arise: the error threshold and the quasispecies. In order to obtain a counterpart of this results for a finite population, we study a Moran model with mutation and selection, and we recover, in a certain asymptotic regime, the error threshold and the quasispecies phenomena. Furthermore, we obtain an explicit formula for the distribution of the quasispecies.

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Variable speed Branching Brownian motion Lisa Hartung, IAM, University of Bonn, Germany

Gaussian processes indexed by trees form an interesting class of correlated random fields where the extremal behaviour can be studied. A rich class of examples are "variable speed" Branching Brownian motions which are defined as ordinary BBM with a time-inhomogeneous variance. In this talk I explain how the convergence of the extremal process for variable speed BBMs is obtained, when the "speed functions" describing the time-inhomogeneous variance lie below their concave hull. The resulting limiting objects turn out to be universal in the sense that they only depend on the slope of the speed function at $0$ and the final time $t$. (joint work with A. Bovier)

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Two dymensional layered systems at the mean field critical temperatureMaria Eulalia Vares, IM-UFRJ, Brazil

I plan to give a short report on recent research in collaboration with L.R. Fontes, D. Marchetti, I. Merola, and E. Presutti. We consider two-dimensional system of Ising spins. On each horizontal line the interaction is given by a ferromagnetic Kac potential at the mean field critical temperature. In addition we have a vertical nearest neighbor ferromagnetic interaction of fixed strength. We then prove that for any positive strength of the vertical interaction the system has a phase transition provided the scale parameter of the Kac interaction is suitably small. Ref. L.R. Fontes, D. Marchetti, I. Merola, E. Presutti, M.E. Vares. "Layered systems at the mean field critical temperature". (arXiv 2015)

OK
Explosion, implosion, and moments of passage times for continuous-time Markov chains : a semi-martingale approachMikhail Menshikov, University of Durham , Department of Mathematical Sciences, United Kingdom

Abstract We establish general theorems quantifying the notion of recurrence— through an estimation of the moments of passage times—for irreducible continuous-time Markov chains on countably infinite state spaces. Sharp conditions of occurrence of the phenomenon of explosion are also obtained. A new phenomenon of implosion is introduced and sharp conditions for its occurrence are proven.

OK
Hydrodynamic Limit, Propagation of Chaos and Large Deviation for a continuous interacting spin systemPatrick Erich Mueller, IAM, University of Bonn, Germany

In this talk I will present results concerning the hydrodynamic limit, the propagation of chaos property, large deviation and the energy landscape for an interacting spin system. Each spin has a fixed spacial position and a continuous spin value that evolves according to a Langevin dynamic. The interaction between two spins depends on their spacial distance, such that local mean field type interactions are covered. Therefore the geometry is highly relevant, especially when the initial distribution is spacial dependent and not iid. By this space dependency this model differs for example from models with mean field interaction. (Joint work with A. Bovier (University of Bonn), D. Ioffe (Technion Haifa))

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Is there a mutation threshold?Rinaldo Schinazi, University of Colorado, USA

We consider a stochastic model for an evolving population. We show that in the presence of genotype extinctions the population dies out for a low mutation probability but may survive for a high mutation probability. This turns upside down the widely held belief that above a certain mutation threshold a population cannot survive.

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## Short Talks (15min)

Statistical mechanics and biological complexity
Adriano Barra, Dipartimento di Fisica, Sapienza Universita di Roma, Italia

Ornstein and Weiss theorem for entrance time and Renyi Entropy

Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions
Daniela Bertacchi, Universita di Milano-Bicocca, Italy

Counting Contours in Trees
Eric Ossami Endo, IME-USP, Brasil

A stochastic individual-based model for immunotherapy of cancer
Hannah Mayer, IAM, University of Bonn, Germany

Geodesic Forests in Last-Passage Percolation
Leandro Pimentel, IM-UFRJ, Brazil

From stochastic, individual-based models to the canonical equation of adaptive dynamics - In one step
Martina Vera Baar, IAM, University of Bonn, Germany

Spectral gap inequality for random walks with long jumps and applications
Milton Jara, IMPA, Brasil

On the shortest way between two cylinders
Rodrigo Lambert, Universidade Federal de Uberlândia, Brazil

Rate functions at zero temperature on countable Markov shifts
Rodrigo Bissacot, Universidade de São Paulo, São Paulo

A dynamical approach to Pirogov-Sinai theory
Santiago Saglietti, Universidad de Buenos Aires, Argentina

Convergence to the invariant measure in the polynuclear growth model (PNG) in the line
Sergio Ivan Lopez Ortega, Universidad Nacional Autonoma de Mexico, Mexico

Scaling limits for the exclusion process with a slow site
Tertuliano Franco, UFBA, Brazil

Statistical mechanics and biological complexityAdriano Barra, Dipartimento di Fisica, Sapienza Universita di Roma, Italia

Statistical mechanics, in particular the know-how of "disordered systems", is playing a maior role in understanding and quantitatively describing biological complexity, ranging from extracellular phenomena as neural and immune networks to intracellular ones as metabolic networks and proteinomics. Aims of this talk is to discuss some behavioral equivalences between the evolution of order parameters in statistical mechanics (i.e. magnetizations) and in chemical kinetics (i.e. concentrations) -a shared langauge for biochemical reactions- so to offer a rationale for the broad consensus and success that statistical mechanics is obtaining in playing its key role in systems biology. As a sideline, I will translate the whole into cybernetical terms, offering an alternative representation of the discussed phenomena in terms of electronic circuits.

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Ornstein and Weiss theorem for entrance time and Renyi EntropyAlejandra Rada Mora, Universidade de São Paulo, Brazil

For ergodic systems with generating partitions, the well known result of Ornstein and Weiss shows that the exponential growth rate of the recurrence time is almost surely equal to the metric entropy. Here we look at the exponential growth rate of entrance times, and show that it equals the entropy, where the convergence is in probability in the product measure. This is however under the assumptions that the limiting entrance times distribution exists almost surely. This condition looks natural in the light of an example by Shields in which the limsup in the exponential growth rate is infinite almost everywhere but where the limiting entrance times do not exist. We then also consider $\phi$-mixing systems and prove a result connecting the R\'enyi entropy to sums over the entrance times orbit segments. Joint work with N. Hadyn, M. Ko, C. Gupta.

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Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactionsDaniela Bertacchi, Universita di Milano-Bicocca, Italy

We introduce spatially explicit stochastic processes to model multispecies host-symbiont interactions. The host environment is static, modeled by the infinite percolation cluster of site percolation. Symbionts evolve on the infinite cluster through contact or voter type interactions, where each host may be infected by a colony of symbionts. In the presence of a single symbiont species, the condition for invasion as a function of the density of the habitat of hosts and the maximal size of the colonies is investigated in details. In the presence of multiple symbiont species, it is proved that the community of symbionts clusters in two dimensions whereas symbiont species may coexist in higher dimensions. Joint work with N.Lanchier and F.Zucca.

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Counting Contours in TreesEric Ossami Endo, IME-USP, Brasil

Using generating functions we are able to compute the exact number of contours of volume $n$ in $d$-regular trees. We use the definition of contour proposed by Babson and Benjamini [1] The problem of counting contours in $\mathbb{Z}^d$ of a fixed volume is one the main steps in the classical Peierls argument and when we want to control polymer expansions. After the Ruelle's bound, Lebowitz and Mazel [3] proved in 1998 that there are between $(C_1d)^{n/2d}$ and $(C_2d)^{64n/d}$ contours of volume $n$ in $\mathbb{Z}^d$ containing a fixed vertex. Using generating functions Balister and Bollobás [2] obtained a nontrivial improvement in 2007, they proved that the number of contours is between $(C_3d)^{n/d}$ and $(C_4d)^{2n/d}$. We show that it is possible to apply the same technique to count the contours in $d$-regular trees and, in this case, we are able to compute the exact number of contours. 1. E. Babson and I. Benjamini, Cut sets and normed cohomology with applications to percolation., Proc. Am. Math. Soc. 127 (1999), no. 2, 589–597. 2. P. N. Balister and B. Bollobás, Counting regions with bounded surface area, Comm. Math. Phys. 273 (2007), no. 2, 305– 315. MR 2318308 (2008m:82039) 3. J. L. Lebowitz and A. E. Mazel, Improved Peierls argument for high-dimensional Ising models, J. Statist. Phys. 90 (1998), no. 3-4, 1051–1059. MR 1616958 (99h:82016)

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A stochastic individual-based model for immunotherapy of cancerHannah Mayer, IAM, University of Bonn, Germany

We propose an extension of a standard stochastic individual-based model in population dynamics which broadens the range of biological applications. The main characteristics of the model are distinction of phenotype and genotype, inclusion of environment-dependent transitions between phenotypes that do not affect the genotype, and the introduction of a competition term which lowers the reproduction rate of an individual in addition to the usual term that increases its death rate. The new setup is illustrated by modelling various phenomena arising in immunotherapy for malignant tumours. On the one hand, we show that the interplay of genetic mutations and phenotypic switches on different timescales as well as the occurrence of metastability phenomena raise new mathematical challenges. On the other hand, we argue why understanding purely stochastic events may help to understand the resistance of tumours to various therapeutic approaches and may have non-trivial consequences on tumour treatment protocols.

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Geodesic Forests in Last-Passage Percolation Leandro Pimentel, IM-UFRJ, Brazil

In this talk we will consider geodesic forests in the last-passage percolation model with exponential weights. We will study problems that are related to the asymptotic location of the root and explain its connection with extrema of random walks. We will also examine the power law behaviour of the height of a tree, its scaling limit, and the relation with the KPZ universality class.

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From stochastic, individual-based models to the canonical equation of adaptive dynamics - In one stepMartina Vera Baar, IAM, University of Bonn, Germany

We consider a model for Darwinian evolution in an asexual population with a large but non-constant populations size characterized by a natural birth rate, a logistic death rate due to age or competition and a probability of mutation at each birth event. In this talk, we focus on the limit of a large population with rare mutations and small mutation effects proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to, e.g. (Champagnat and Méléard, 2011), we study the three limits simultaneously, subject to conditions that ensure that the time scale of birth and death events remains separated from that of successful mutational events. The fact that the mutational effect tends to zero with the population size slows down the dynamics of the microscopic system and leads to serious technical difficulties that require the use of completely different methods than those used in previous work.

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Spectral gap inequality for random walks with long jumps and applicationsMilton Jara, IMPA, Brasil

We consider a random walk on the integer lattice with jump rate belonging to the normal domain of attraction of an stable law. We obtain sharp lower and upper bounds for the first non-zero eigenvalue of the walk restricted to a finite box of increasing size. The proof is based on a general renormalization procedure, which can be applied to obtain similar bounds for various stochastic interacting particle systems with long jumps, like the exclusion process, the zero-range process and the Ginzburg-Landau process.

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On the shortest way between two cylindersRodrigo Lambert, Universidade Federal de Uberlândia, Brazil

Let us consider the minimum number of steps spent to a process starting at a inicial condition $y_1^n$ to reach the target-condition $x_1^n$. We call it the shortest way between two cylinders. If the inicial condition and the target-condition are generated by independent ergodic with positive entropy measures, we prove a "strong law of large numbers" for this quantity. If the measures have the additional Pittel condition, we prove a Large Deviation Principle. Finally, if the measure are beta-mixing, we prove a convergence in distribution for a re-scaled version of this quantity.

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Rate functions at zero temperature on countable Markov shiftsRodrigo Bissacot, Universidade de São Paulo, São Paulo

For the class of topologically mixing countable Markov shifts $\sum_A(N)$ with the BIP property over the alphabet N, given a Walters potential $f : \sum_A(N) \rightarrow R$ with Gurevich finite pressure we are able to find the rate function associated to a large deviation principle for the family of Gibbs states $(\mu \beta ) \beta >$1 [$\mu \beta$ is the Gibbs measure associated to the potential $\beta f$] when $\beta$ goes to infinity. As a corollary, we obtain a new proof for the same principle in the case of topologically mixing subshifts over a finite alphabet and a proof of the existence of maximizing measures for the class of Walters potentials.

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A dynamical approach to Pirogov-Sinai theorySantiago Saglietti, Universidad de Buenos Aires, Argentina

We combine the loss-network dynamics introduced in [1] with the framework of Pirogov-Sinai theory to obtain new proofs of the stability results given by this theory, which replace the traditional use of cluster expansions with tools from the theory of stochastic processes. This dynamical approach allows us to enlarge the standard range of stability in some models while also yielding a perfect simulation algorithm for the infinite-volume stable phases of systems in the non-uniqueness regime. [1] Fernandez, Roberto, Ferrari, Pablo A., Garcia, Nancy L. Loss network representation of Peierls contours. Ann. Probab. 29 (2001), no. 2, 902-937.

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Convergence to the invariant measure in the polynuclear growth model (PNG) in the lineSergio Ivan Lopez Ortega, Universidad Nacional Autonoma de Mexico, Mexico

We will talk about the PNG model in the line considered by Prahofer and Spohn (2004) . This growth model, which is in the KPZ class, is closely related to the Hammersley process. In this work we show uniqueness of the stationary measure by the use of similar techniques to those applied by Mountford and Prabhakar (1995) to prove uniqueness of stationary measure for the - /M/1 queue operation. Joint work with Ines Armendariz

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Scaling limits for the exclusion process with a slow siteTertuliano Franco, UFBA, Brazil

We consider the symmetric simple exclusion processes with a slow site in the discrete torus with n sites. In this model, particles perform nearest- neighbor symmetric random walks with jump rates everywhere equal to one, except at one particular site, the slow site, where the jump rate of entering that site is equal to one, but the jump rate of leaving that site is given by a parameter $g(n)$, where $n$ is the scaling parameter. We consider two different strengths $g$. In the first, both the hydrodynamic behavior and equilibrium fluctuations are driven by the heat equation with periodic boundary conditions, while in the second they are driven by the heat equation with Neumann's boundary conditions. We therefore establish a phase transition. The critical behavior/strength remains open. Joint work with G. Schutz and P. Gon\c calves.

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## Posters

The posters will be pasted on the wall with tape. Its dimensions must fit in a space of 120 cm high by 90 cm wide.

Poster Session 1 - Monday
An extension to the theory of weak convergence with application for coalescent processes
Anatoli Iambartsev, IME-USP, Brasil

Soft Local Times
Caio T M Alves, Unicamp, Brasil

Dynamic uniqueness for chains of infinite order
Christophe Gallesco, IMECC-UNICAMP, Brazil

Convergence to a transformation of the Brownian Web for a family of Poissonian trees.
Cristian Favio Coletti, UFABC, Brasil

Subcritical phase for the Boolean model of Percolation on Nilpotent groups.

Monitoring tweets: a quantitative analysis
Geraldine Bosco, Universidade de São Paulo, Brasil

Cut off phenomenon for $d$-dimensional Itô diffusions: the generic case.
Gerardo Barrera Vargas, IMPA, Brasil

Central Limit Theorem for the self-repelling random walk with directed edges
Glauco Valle da Silva Coelho, IM-UFRJ, Brasil

Hockey Stick Pattern" of Pascal's Triangle and Random Experiments: Do they have anything in common?
Iesus Diniz, UFRN, Brasil

Uniqueness vs. non-uniqueness in complete connections with modified majority rules.
Jeanne Dias, UFMG, Brasil

Percolation on random causal triangulation
José Cerda Hernández, IMECC-UNICAMP, Brazil

Open Markov chain models with times series feeds
José Moniz Fernandes, Universidade de Cabo Verde, Cabo Verde

About Mean Convergence Time of Inhomogeneous Simulated Annealing with Elitism.
Juan Cruz, UFRN, brasil

A spatial stochastic model for a two-stages innovation diffusion
Karina Bindandi Emboaba de Oliveira, USP-ICMC/UFSCar, Brasil

A New Discrete-type Aging Property and Applications
Leandro Ferreira, IME-USP, Brasil

Generalized Random directed forest process: Convergence to the Bronian Web
Leonel Zuaznabar, IM-UFRJ, Brasil

Poster Session 2 - Tuesday
Dynamical systems with heavy-tailed random parameters
Marina Vachkovskaia, IMECC-UNICAMP, Brasil

Image Denoising using Stochastic Differential Equations
Michel Luna Ccora, Universidade Federal Fluminense , Brasil

Potential well spectrum for renewal process

A New Discrete-type Aging Property and Applications
Nikolai Kolev, IME-USP, Brasil

TBA
Pablo Rodríguez, ICMC-USP, São carlos

Thermodynamical Stability of Many-Body Lattice Quantum Hamiltonians with Multiparticle Interactions
Paulo Afonso Faria da Veiga, ICMC-USP, Brazil

Application of Stochastic Calculus in Finance
Rafael Console, ICMC-USP, Brasil

Coexistence in the two-type Richardson model
Rangel Baldasso, IMPA, Brasil

TBA
Renato Gava, UFSCar, Brazil

An Euler-Maruyama-type approach for the Cox-Ingersoll-Ross process
Ricardo Ferreira, USP-ICMC/UFSCar, Brasil

Exact invariant measures: How the strength of measure settles the intensity of chaos
Roberto Venegeroles, UFABC, Brazil

Parametric and semi-parametric specifications of multi-state models.
Robson Jose Mariano Machado, University College London, United Kingdom

Cliques in Random Graphs
Rodrigo Ribeiro, UFMG, Brasil

Shannon Entropy and Mutual Information in Log Asymmetric Distributions With Normal Kernel
Roger Silva, UFMG, Brasil

Advances on the Late Arrivals Problem
Sokol Ndreca, UFMG , Brasil

Effect of stochastic transition in the fundamental diagram of traffic flowAdriano Siqueira, EEL-USP, Brasil

In this work, we propose a stochastic model to the fundamental diagram of traffic flow with minimal number of parameters. Our approach is based on a mesoscopic view of the traffic system in terms of the dynamics of vehicle speed transitions. A key feature of the present approach is the use of stochastic differential equations which makes it possible to describe not only the flow - density, called fundamental diagram in the traffic engineering, but also the variability. The model was tested to describe the flow of vehicles on the "May 23" highway in the Brazilian city of São Paulo, where both the fundamental diagram and its variance are reasonably reproduced. Despite its simplicity, we argue que the current model provides an alternative description for the fundamental diagram in the study of traffic flow.

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Colonization and CollapseAlejandro Roldan, IME-USP, Brasil

Many species (such as ants) live in colonies that thrive for a while and then collapse. Upon collapse very few individuals survive. The survivors start new colonies at other sites that thrive until they collapse, and so on. We introduce a spatial stochastic process on graphs for modeling such type of population dynamic. This is a joint work with F. Machado and R. Schinazi.

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An extension to the theory of weak convergence with application for coalescent processesAnatoli Iambartsev, IME-USP, Brasil

The goal of this note is to prove an extension for so-called density dependent population processes. Slightly modified proofs from the book of S.N.Ethier and T.Kurtz Markov processes: Characterization and Convergence'' provide the low of large number and central limit theorem for such processes. Under the new conditions it is possible apply the powerful theory of convergence (developed in the above book) for a wide class of coalescent processes.

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Soft Local TimesCaio T M Alves, Unicamp, Brasil

Alain-Sol Sznitman introduziu em \cite{sznitman} o modelo dos entrelaçamentos aleatórios em $\mathbb{Z}^d$ para explicar a figura local que uma trajetória de passeio aleatório simples no toro $d$-dimensional $\mathbb{Z}^d/n\mathbb{Z}^d$ descreve em um subconjunto pequeno. O modelo trata-se de uma nuvem poissoniana de trajetórias de passeios aleatórios simples em $\mathbb{Z}^d$, com $d\geq 3$. A quantidade de trajetórias é controlada por um parâmetro real $u$, a união de todas as trajetórias é denotada por $\mathcal{I}^u$, os chamados entrelaçamentos no nível $u$, e seu complemento, $\mathbb{Z}^d\setminus \mathcal{I}^u$, é denotado por $\mathcal{V}^u$, o chamado conjunto vacante no nível $u$.

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A stochastic modeling for evolution with mutationCarolina Grejo, IME-USP, Brasil

We propose a stochastic model for evolution with mutation. The model evolves on a two-dimensional oriented tree. At time 0 the root is occupied only by one species of type 1. Each type waits a exponential time, with rate $\lambda$, to give birth a one new individual. With probability 1-r the new individual has the same type that its mother, and will occupy the same vertex on the tree that her, and with probability r, the new individual is a mutation, so it has the same type that its species mother plus 1. If it is the first mutation that type, it will occupy the vertex to the left, below the mother species' vertex, if it is the second mutation, it will occupy the vertex to the right. After the appearance of each type, they evolve independently of the others. Note that at each level the tree is occupied by just one type. Let the root be the level 1, the first two vertex under the tree root the level 2, and so one. Death occurs following a Poisson process with rate 1 (PPP1) that evolves independently of the birth process. At each PPP1 mark, one level will be extinct of the system. At the first mark will be the level one, on the second mark will be level 2, and so one. We prove that for each r fixed and for all branchs, there is a phase transition. If $\lambda \leq \frac{1}{r}$ there is extinction along all branchs and if $\lambda >\frac{1+\sqrt{2}}{r}$ with positive probability all branchs will survive.

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Fluctuation bounds for entropy and entropy production estimation in Gibbs measuresCesar Maldonado, Centro de Modelamiento Matemático. U.Chile., Chile

We study the fluctuation properties of entropy and entropy production estimators for one-dimensional Gibbs measures. The bounds we obtain are valid for every n, being n the sample length. Thus, they are interesting from the practical point of view. For entropy estimators we study the so-called “Plug-In” estimator, the conditional entropy and the waiting time. In the case of the entropy production estimators we study the fluctuations of the waiting time and the hitting time as estimators of the mean entropy production. Our bounds are based on concentration results and the exponential law of the measure of a given cylinder. For the sake of completeness I review also the asymptotic results on this estimators in both cases.

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Dynamic uniqueness for chains of infinite orderChristophe Gallesco, IMECC-UNICAMP, Brazil

We say that a kernel exhibits dynamic uniqueness if all the chains starting from a fixed past coincide in the future tail $\sigma$-algebra, otherwise the kernel exhibits dynamic phase transition. We characterize dynamic uniqueness/phase transition by proving several equivalent conditions. In particular, we prove that dynamic uniqueness is equivalent to convergence in total variation distance of all the chains starting from different pasts. We also study the relationship between our definition of uniqueness and the $\ell^2$ criteria for the uniqueness of $g$-measures. We prove that the Bramson-Kalikow and Hulse models exhibit dynamic uniqueness if and only if the kernel is in $\ell^2$. Finally, we prove that a $g$-measure $P$ is weak Bernoulli (or, equivalently, $\beta$-mixing) if and only if $g$ exhibits dynamic uniqueness for $P$-a.e. pasts, generalizing several results in the literature.

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From a Reaction Diffusion model to Study of a Stochastic Differential EquationConrado Da Costa, IMPA, Br

We explore martingales techniques to prove tightness of a reaction diffusion model and show that the limit probability measures satisfy a Singular Stochastic Differential equation. This shall be enough to prove convergence of the family to the single limit, once we show that the martingale problem of Stroock and Varadhan is well posed for this particular SSDE. keywords: reaction diffusion model, particle system, martingale approach, Singular Stochastic Differential equation

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Convergence to a transformation of the Brownian Web for a family of Poissonian trees.Cristian Favio Coletti, UFABC, Brasil

We introduce a system of one-dimensional coalescing random paths starting at the space-time points of a homogeneous Poisson point process in $\mathbb{R} \times \{0\}$ which are constructed as a function of a family $(\Lambda_n)_{n \in \mathbb{N}}$ of Poisson point processes. We show that under diffusive scaling this system converges in distribution to a continuous mapping of the Brownian Web. Joint work with Leon A. Valencia Henao (UdeA, Colombia).

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Phase Transition for some Ellipses Percolation Models on $\mathbb{R}^2$Daniel Borges, IMPA, Brasil

Percolation theory is a very active area of study in probability. Since many problems about Bernoulli percolation have already been answered, the research focus has shifted to more complex models, which have long range dependencies. I will introduce a model of continuous percolation with random ellipses in $\mathbb{R}^2$, inspired by the paper "Percolation in the vacant set of Poisson cylinders" of Tykesson and Windisch. The model is a generalization of the random ellipses that appear naturally in their article, through the introduction of a new parameter which controls the (random) size of the ellipses' major axes. I prove a phase transition for percolation of the vacant set, the set obtained after removing all random ellipses .

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Subcritical phase for the Boolean model of Percolation on Nilpotent groups.Daniel Machado, UFABC, Brazil

We consider the discrete Boolean model of percolation on nilpotent groups. Informally, we may describe the Boolean model of percolation on discrete structure as follows. Consider a simple point process X in some polish, locally compact metric space $(\Gamma, d)$. Then, at each point of X , center a ball of random radius. Assume that the radius are independent, identically distributed and independent of X. We provide sufficient conditions guarranteeing the existence of a subcritical phase.

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Random Interlacements in one dimensionDarcy Camargo, IMECC-UNICAMP , Brasil

We define a one dimensional version of capacity and then respective random interlacements process. We also show that a conditional random walk on the ring graph for a fixed time converges to this random interlacement process.

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Some applications of Russo's formula for random interlacementsDiego Fernando de Bernardini, IMECC-UNICAMP, Brasil

Russo's formula for random interlacements establishes some expressions for the derivative of the probability of an increasing event in the random interlacements process. In this work we will discuss some applications of these expressions. For example, we obtained upper bounds on the expected number of plus-pivotal trajectories for an increasing event in the random interlacements model. Also, when we can obtain explicit expressions for the probability of an increasing event, we see that it is possible to establish an explicit expression for the expected number of trajectories in the interlacements process restricted to the set where the event is supported, conditioned on the occurrence of that event. This is a joint work with Serguei Popov, and it is based on our recent paper, entitled "Russo's Formula for Random Interlacements", which was accepted for publication in the Journal of Statistical Physics. This work was supported by São Paulo Research Foundation - FAPESP (Grant number 2014/14323-9).

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Connectivity threshold for the Erdos-Renyi random graphElizbeth Chipa Bedia, USP-ICMC/UFSCar, Brasil

In this work we consider the Erdos-Rényi random graph model, which is defined taking a set of n vertices and connecting each pair of them with probability $p = \lambda /n$ for some $1 < \lambda < \infty$ such that $0 < p < 1$, independently of the other pairs of vertices. We study the connectivity threshold for this model.

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Monitoring tweets: a quantitative analysisGeraldine Bosco, Universidade de São Paulo, Brasil

Extracted public available social media data has been extensively used to monitor public health status, to measure information diffusion and to track the effectiveness of health marketing campaigns. In this work we deal with some important research questions: What information related to seasonal flu vaccination in Brazil can be learned from monitoring microblogs like twitter? What are the positive and negative sentiments associated with this specific campaign? Is there a pattern to describe sentiments changes? We have analyzed quantitatively these questions and have also engaged into an interesting comparison between two disease prevention campaigns, the 2014 seasonal flu vaccination in Brasil, and the 2009H1N1 outbreak in USA. We have collected tweets during the 2014 seasonal flu vaccination in Brasil categorization analysis of these tweets in three major axis: content, qualifier and links, is described and the comparison with the data collected during the H1N1 outbreak in USA is also disclosed. We have proposed new categories to sentiment analysis beyond existing standards. Preliminarily, we have used statistical tests and naïve Bayes classifier to analyzing the data, however, we plan to use complex network to understand some relationship between some significant expressions.

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Cut off phenomenon for $d$-dimensional Itô diffusions: the generic case.Gerardo Barrera Vargas, IMPA, Brasil

We study the cut-off phenomenon for a family of stochastic perturbations of a dynamical system in $\mathbb{R}^d$. We will focus in a semi-flow of a deterministic differential equation with an unique single attractor which is perturbed by an standar Brownian motion of small variance. Under suitable hypothesis on the vector field we will prove that the family of perturbed stochastic differential equations present a profile cut-off phenomenon.

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Central Limit Theorem for the self-repelling random walk with directed edges Glauco Valle da Silva Coelho, IM-UFRJ, Brasil

We prove the weak convergence of the self repelling random walk with directed edges under diffusive scaling to a uniform distribution.

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Simple models of strings of characters with infinite alphabetHelder Alan Rojas Molina, IME-USP, SAO PAULO

abstract: We consider the a family of Markov chains defined on the sequences of characters (strings, or words) with infinite alphabet. For some examples inspired by the models of high frequency trading we obtain a conditions for ergodicity, transience and null-recurrence. In order to prove this we use the construction of Lyapunov functions techniques.

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Extended Marshal-Olkin Model with a Positive Mass on Arbitrary LineHugo Brango, IME-USP, Brasil

We suggest an extension of the classical bivariate Marshall-Olkin model in two directions. First, we relax the independence assumption between "individual shocks" and second, we consider a version with a positive mass concentrated on arbitrary line. Related characterizations and applications of the new model will be discussed.

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Hockey Stick Pattern" of Pascal's Triangle and Random Experiments: Do they have anything in common?Iesus Diniz, UFRN, Brasil

The most common proofs of the "Hockey Stick Pattern" of Pascal´s triangle, formally given in (\ref{TeoCol}), are obtained by the principle of mathematical induction~\cite{Lovasz}, by combinatorial argument~\cite{Eletronico}, or recursively from Stifel's~\cite{Brualdi}. In this short article, we prove the "Hockey Stick Pattern" of Pascal´s triangle by using three different probabilistic arguments. In contrast with inductive proofs, which require us to know the formula we want to prove in advance, a probabilistic argument builds a model from which the formula is obtained (without the need for knowing it in advance). In addition, alternative proofs of any mathematical statement are always desirable from both a pedagogical and a theoretical point of view; specifically, they stress the connection between distinct areas of mathematics. In particular, our paper is an example of the application of an elementary probabilistic tool to prove a result in another area: combinatorics and number theory.

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Uniqueness vs. non-uniqueness in complete connections with modified majority rules.Jeanne Dias, UFMG, Brasil

We consider a class of chains with complete connections inspired by the one of Berger, Hoffman and Sidoravicius. We show that if the majority rule used to fix the dependence on the past is replaced with a smooth function (differentiable at the origin), the system looses its memory.

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Percolation on random causal triangulationJosé Cerda Hernández, IMECC-UNICAMP, Brazil

In this work we study bond percolation on random causal triangulation. We show that the phase transition is non-trivial and we compute first bounds for the critical value. Clearly, the critical value depends sensitively on the nature of the underlying graph, but the critical value is shown to be constant a.s.

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Open Markov chain models with times series feedsJosé Moniz Fernandes, Universidade de Cabo Verde, Cabo Verde

In [1] an open Markov chain model - that is, with a random entering flow of new members in the population (see also [2] and [3] for details on the methods used)- was applied a consumption credit portfolio with five risk classes, in order to estimate the global portfolio risk spread. In this work - aiming at prediction results - we impose a time series additional structure on the inflow of new members. We consider two different approaches: we fit an ARIMA process to the monthly entrance inflow numbers and, in the second approach, after fitting a sigmoid function to the inflow numbers, we fit a SARMA process to the residuals of the sigmoidal function fitting. In the previsiouly mentioned consumption credit portfolio context we compare, by simulation, the spread results obtained with these two approaches. As [1], our traitment is illustrated with real data from a Cape verdean bank. [1] Esquível, M.L., Guerreiro, R.G. and Fernandes, J.M. (2014), On the Evolution and Asymptotic Analysis of Open Markov Populations Subjected to Periodic Reclassifications, with Stochastic Models, 30(3), 365-389. DOI:10.1080/15326349.2014.912947. [2] Guerreiro, G.R.; Mexia, J.T. ; Miguens, M.F. (2014), Statistical Approach for Open Bonus Systems, ASTIN Bulletin, 44(1), 63-83. [3] Guerreiro, G.R.; Mexia, J.T. ; Miguens, M.F. (2012), Stable Distributions for Open Populations Subject to Periodical Re-classifications, Journal of Statistical Theory and Practice, 6(4), 621-635. keywords: Finance and Banking, Financial Modelling, Stochastic Models.

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About Mean Convergence Time of Inhomogeneous Simulated Annealing with Elitism.Juan Cruz, UFRN, brasil

The theoretical study of Simulated Annealing (SA) has focused mainly on establishing their convergence in probability and almost surely to the global optimum. In this paper we establish sufficient conditions for the mean convergence time of the inhomogeneous (SA) with elitism of a given function f being finite.

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A spatial stochastic model for a two-stages innovation diffusionKarina Bindandi Emboaba de Oliveira, USP-ICMC/UFSCar, Brasil

We propose and study a spatial stochastic model describing a process of awareness, evaluation and decision-making by agents on the $d$-dimensional integer lattice. Each agent may be in any of the three states belonging to the set $\{0, 1, 2\}$. In this model $0$ stands for ignorants, $1$ for aware and $2$ for adopters. Aware and adopters tell about a new product innovation to any of its (nearest) ignorant neighbors at rate $\lambda$. At rate $\alpha$ an aware becomes an adopter due to the influence of (nearest) adopters neighbors. Finally, aware and adopters forget the information about the new product at rate one. The purpose of this work is to analyze the influence of the parameters on the qualitative behavior of the process. More precisely, we study sufficient conditions under which the innovation diffusion (and adoption) either becomes extinct or propagates through the population with positive probability. This is a joint work with Cristian Coletti (UFABC) and Pablo Rodríguez (ICMC-USP).

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A New Discrete-type Aging Property and ApplicationsLeandro Ferreira, IME-USP, Brasil

Let us consider two non-negative integer valued random variables with their survival functions. A discrete version of bivariate lack of memory property (BLMP) preserves the distribution of the random vector and its residual lifetime vector independent of the "time". We will present a new aging version that conserves the Sibuya dependence function of the random vectors Such a version includes as a particular case the discrete BLMP. Related charcterizations will be reported, along with many geometric-type bivariate geometric distributions possessing such a property. Reliability and engineering applications will be discussed.

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Generalized Random directed forest process: Convergence to the Bronian WebLeonel Zuaznabar, IM-UFRJ, Brasil

A natural generalization of the process studied by Rahul Roy, Kumarjit Saha and Anish Sarkar named Random directed forest is present and some results in the way to prove the convergence to the Brownian web.

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Projective distance and $g-$measures.Liliana Trejo, IF-UASLP, México

We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to the $\bar{d}$-distance and we prove that in general they are not comparable. Our projective distance is inspired on Hilbert’s projective metric, and in the framework of $g$-measures, it allows to assess the continuity of the entropy at $g$-measures satisfying uniqueness. It also allows to relate the speed of convergence and the regularity of sequences of locally finite $g$-functions, to the preservation at the limit, of certain ergodic properties for the associate $g$-measures.

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Uniqueness region in a ferromagnetic Ising model with periodical external fieldManuel González Navarrete, Universidade de São Paulo, Brasil

We consider the ferromagnetic Ising model with a cell-board external field, studied in \cite{GPY}. Particularly, we characterize the low-temperature diagram in the region with a unique ground state at $T=0$. The external field is associated to a cell-board configuration, with positive and negative spin values alternating in rectangular cells of sides $L_1\times L_2$ sites, such that the total value of the external field is zero. If the external field takes two values: $h$ and $-h$, where $h>0$, the uniqueness region is $h>h_c :=\frac{2J}{L_1}+ \frac{2J}{L_2}$, where $J$ is an interaction constant. This work is a continuation of \cite{GPY}, where was proved a phase transition at low temperature in $h < h_c$, using the property of reflection positivity (RP). We will use the methods of disagreement percolation \cite{vdB} and cluster expansion \cite{FP}. \begin{thebibliography}{99} \bibitem{vdB}{van den Berg, J., Maes, C.} \newblock{\it Disagreement percolation in the study of Markov fields}. \newblock Ann. Probab. \textbf{22}: 749-763 (1994). \bibitem{FP}{Fernandez, R., Procacci, A.} \newblock{\it Cluster expansion for abstract polymer models.New bounds from an old approach}. \newblock Commun. in Math. Phys. \textbf{274}(1): 123-140 (2007). \bibitem{GPY}{Gonzalez Navarrete, M., Pechersky, E., Yambartsev, A.}: \newblock {\it Phase transition in ferromagnetic Ising model with a cell-board external field}. \newblock arXiv:1411.7739. \end{thebibliography}

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Dynamical systems with heavy-tailed random parametersMarina Vachkovskaia, IMECC-UNICAMP, Brasil

Motivated by the study of the time evolution of random dynamical systems arising in a vast variety of domains --- ranging from physics to ecology ---, we establish conditions for the occurrence of a non-trivial asymptotic behaviour for these systems in the absence of ellipticity condition. The problem is tackled by mapping the random dynamical systems into Markov chains on $\mathbb R_+$ with heavy-tailed innovation and then using powerful methods stemming from Lyapunov functions to map the resulting Markov chains into positive semi-martingales. Joint work with V. Belitsky, D. Petritis, M. Menshikov.

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Image Denoising using Stochastic Differential EquationsMichel Luna Ccora, Universidade Federal Fluminense , Brasil

There exist numerous approaches to image denoising based on filtering. Stochastic methods have been proposed to avoid dependence of the initial configuration in such an algorithms. In those approaches, energy is interpreted as a Hamiltonian on the space of all possible configurations, which is minimized by a Metropolis-Hastings algorithm (MHa). In this monograph we intend to study a restoration algorithm recently proposed by Descombes-Zhizhina, based on stochastic relaxation and annealing in which the relaxation process is given by an Ito diffusion with a Gibbs stationary measure. This approach turns to be advantageous, compared with MHa, when a small number of iterations are performed.

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Potential well spectrum for renewal processMiguel Abadi, IME-USP, Brasil

The potential well of a state can be interpreted physically as the energy that a stationary process needs to leave the state. We prove that for discrete time renewal processes, the potential well is the right scaling for the hitting and return time distributions of the state. We further detail the potential well spectrum of these processes by giving a complete classification of the states according to their potential well.

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A New Discrete-type Aging Property and ApplicationsNikolai Kolev, IME-USP, Brasil

Let us consider two non-negative integer valued random variables with their survival functions. A discrete version of bivariate lack of memory property (BLMP) preserves the distribution of the random vector and its residual lifetime vector independent of the "time". We will present a new aging version that conserves the Sibuya dependence function of the random vectors Such a version includes as a particular case the discrete BLMP. Related charcterizations will be reported, along with many geometric-type bivariate geometric distributions possessing such a property. Reliability and engineering applications will be discussed.

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TBAPablo Rodríguez, ICMC-USP, São carlos

TBA

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Thermodynamical Stability of Many-Body Lattice Quantum Hamiltonians with Multiparticle InteractionsPaulo Afonso Faria da Veiga, ICMC-USP, Brazil

We analyze a quantum system of $N$ identical particles of mass $m$, in the lattice $\mathbb{Z}^d$. The particles interact via local two and three-body interactions. We give conditions under which the system verifies thermodynamical stability, that means, conditions under which the system Hamiltonian $H_N$ possess a lower linear bound in $N$.

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Application of Stochastic Calculus in FinanceRafael Console, ICMC-USP, Brasil

In this work we study the basis of stochastic calculus, such as Brownian motion, stochastic integral, Ito's formula, stochastic differential equation and Girsanov's theorem needed to understand de Black and Scholes model and the strategy of pricing and hedging an option. We also see that the assumptions of constant volatility and return's normality sometimes aren't satisfied.

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Coexistence in the two-type Richardson modelRangel Baldasso, IMPA, Brasil

The two-type Richardson is a modification of first passage percolation, where two competing species fight for survival. One interesting question about this model is whether there is a positive probability that each one of the species occupy infinitely many sites. It is conjectured that mutual infinite survival is possible if and only if the two species evolve with the same speed. It is known that in the later case mutual survive is possible. The second part of the conjecture has not been completely proved yet. We present some partial results.

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TBARenato Gava, UFSCar, Brazil

TBA

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An Euler-Maruyama-type approach for the Cox-Ingersoll-Ross processRicardo Ferreira, USP-ICMC/UFSCar, Brasil

The Cox-Ingersoll-Ross process was originally proposed by John C. Cox, Jonathan E. Ingersoll Jr. and Stephen A. Ross in 1985. Nowadays, this process is widely used in financial modelling, e.g. as a model for short-time interest rates or as volatility process in the Heston model. The stochastic differential equation (SDE) which defines this model does not have closed form solution, so we need to approximate the process by a numerical method. In the literature, several numerical approximations has been proposed based in interval discretization. In this work, we approximate the CIR process by Euler-Maruyama-type method based in random discretization proposed by Leão e Ohashi (2013) under Feller condition. In this context, we obtain an exponential convergence order for this approximation and we use Monte Carlo techniques to compare the numerical results with theoretical values.

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Exact invariant measures: How the strength of measure settles the intensity of chaosRoberto Venegeroles, UFABC, Brazil

The aim of this work is to show how extracting dynamical behavior and ergodic properties from deterministic chaos with the assistance of exact invariant measures. On the one hand, we provide an approach to deal with the inverse problem of finding nonlinear interval maps from a given invariant measure. Then, we show how to identify ergodic properties by means of transitions along the phase space via exact measures. On the other hand, we discuss quantitatively how in finite measures imply maps having subexponential Lyapunov instability (weakly chaotic), as opposed to finite measure ergodic maps, that are fully chaotic. In addition, we provide general solutions of maps for which infinite invariant measures are exactly known throughout the interval (a demand from this field). Finally, we give a simple proof that infinite measure implies universal Mittag-Leffler statistics of observables, rather than narrow distributions typically observed in finite measure ergodic maps.

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Parametric and semi-parametric specifications of multi-state models.Robson Jose Mariano Machado, University College London, United Kingdom

Continuous-time multi-state models are formulated to investigate transition intensities for discrete states over time. For the intensities, various specifications are considered using parametric and semi-parametric expressions. Estimation is performed by using maximum likelihood. The models are illustrated with data describing the development of cardiac allograft vasculopathy, a deterioration of the arterial walls, in heart transplantation patients.

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Cliques in Random GraphsRodrigo Ribeiro, UFMG, Brasil

In this work we will study some properties of graphs generated by a random process with preferential attachment rules. In this specific model, at each step, we decide according to Bernoulli random variables, whether we add a new vertex or only a new edge to the previous graph. We look for results concerning concentration of measure of the degrees and a possible region with agglomeration, i.e., the appearance of complete subgraphs in the graph generated by this kind of dynamics.

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Shannon Entropy and Mutual Information in Log Asymmetric Distributions With Normal KernelRoger Silva, UFMG, Brasil

In this work we provide the Shannon entropy for a wide class of skew and log skew multivariate distributions with normal kernel, namely the Canonical Fundamental Skew Normal and Log Canonical Fundamental Skew Normal distributions and express them in terms of the entropy of a Multivariate Normal distribution. In addition we obtain the information index and the Kullback-Leibler measure for these distributions.

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Advances on the Late Arrivals ProblemSokol Ndreca, UFMG , Brasil

We study a discrete time queueing system where deterministic arrivals have i.i.d.\ exponential delays $\xi_{i}$. The standard deviation $\sigma$ of the delay is finite, but much larger than the deterministic unit interarrival time. We describe the model as a bivariate Markov chain and focus on the joint equilibrium distribution, proving that the latter decays super-exponentially fast in the quarter plane. Finally, we discuss the numerical computation of the stationary distribution, showing the effectiveness of a simple approximation scheme in a wide region of the parameters. The model, motivated by air and railway traffic, was proposed many decades ago by Kendall with the name of late arrivals problem'', but no solution has been found so far. Joint work with Carlo Lancia, Gianluca Guadagni and Benedetto Scoppola.

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The cone percolation model on spherically symmetric trees and its variationsValdivino Vargas Júnior, UFG, Brasil

We consider a percolation process which allows us to associate the dynamic activation on the set of vertices to a discrete rumor process. Individuals become spreaders as soon as they heard about the rumor. Next time, they propagate the rumor within their radius of influence and immediately become stiflers. Here we focus on homogeneous, periodic and spherically symmetric trees in a process which considers general positive random variables and their distributions for radius of influences.

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