## Seminários de Probabilidade 2019

##### Seminários de Probabilidade do Departamento de Métodos Estatísticos - Instituto de Matemática - UFRJ

**Seminários de Probabilidade de 2019**

**Coordenação:** Professora Maria Eulalia Vares

As palestras ocorrerem na sala B106-b nas segundas-feiras às 15h30, a menos de algumas exceções devidamente indicadas.

###### Lista Completa

###### 25/11

Cell lineage data comes from single-cell transcriptomics and it is used to recover the evolutionary path of cells in a given environment. The different evolutionary stages of the cells can be probabilistically described by distinct components in a mixture model. This work proposes a Bayesian dependent mixture model where the dependence on the components of the mixture explicitly incorporates the biological structure that characterizes cell lineage applications. We use a random tree structure (Minimum Spanning Tree) not only to explain the snapshot in the latent space of the continuous development of cells from its initial stage into mature differentiated cells, but also to model the dependence structure between the clusters of cells. Regularization is incorporated in the form of a prior penalization on trees with too many nodes or with redundant edges. Consequently, the model assumes the partition of cells to depend on the lineage structure, which is more biologically reasonable then the usual multistep approach in which partitions are estimated disregarding the underlying tree structure that characterizes cell lineage data. We are able to provide full inference (with uncertainty captured by the posterior samples obtained through MCMC) on the clusters of cells (including number of clusters), on the underlying tree structure and also on pseudotimes.

Authors: Zanini, C. T. P, Paulon, G., Mueller, P.

###### 11/11

We will discuss the two-dimensional simple random walk conditioned on never hitting the origin, which is, formally speaking, the Doob’s h-transform of the simple random walk with respect to the potential kernel. This random walk is the main building-block of the construction of random interlacements on the plane introduced by Comets, Popov and Vachkovskaia. However, this walk has become an interesting object on its own. To justify this claim we present a few of its properties, citing some of the current literature and presenting the results of a recent joint work with Serguei Popov (UNICAMP) and Leonardo Rolla (UBA/NYU Shanghai).

###### 07/10

We propose the notion of replicas in the context of discrete choices and introduce axioms that support which we call the Luce model with replicas. Unlike other relations proposed in the literature that can deal with the duplicates problem, ours entails replicas as a combination of duplicates and stochastically perfect substitutes, which induces a partition of the entire set of alternatives into endogeneous nests of replicas. Our model is less restrictive than Luce.s model and more parsimonious than the available models that may deal with the violation of the constant-ratio rule anticipated by Debreu (1960).

###### 23/09

In this talk we will review the recent developments on Functional Itô calculus, FITO in short. Created (or discovered) by Bruno Dupire and published in a seminal paper in 2009, this calculus is a generalization of Itô’s classical theory and allows us to examine models where the history of certain factors plays an important role. We will present the general theory and survey the theoretical unfolding of FITO. As an application, we will show how this theory allows us to consider stochastic control problems with path-dependence influence of the control in the dynamics of the state process.

###### 16/09

In this presentation, we will discuss the existence of a local solution to the equation

$$

\partial X = -(-\Delta)^{1/2} X – \sinh(\gamma X) + \xi,

$$

where $(-\Delta)^{1/2}$ is the half-laplacian, and $\xi$ is the space-time white noise. As the solution is not point-wise welldefined function, we will have to define the meaning of $\sinh(\gamma X)$. We will also discuss the basic ideas between da Pratto Debusche approach to non-linear SPDE’s and more modern techniques, such as regularity structures.

###### 09/09

A teoria de filas é um ramo da probabilidade que estuda sistemas de atendimento onde temos um processo estocástico que descreve a chegada de clientes de forma ordenada a uma unidade de atendimento, formando-se filas de espera, e um outro que descreve o tempo atribuído a cada serviço. Nessa apresentação iremos estudar propriedades assimptóticas de um sistema onde clientes passam por uma série de filas em ordem, e os tempos de atendimento são dados por movimentos Brownianos.

###### 26/08

O presente trabalho apresenta uma abordagem integrada para o processo de discriminação de preços para assinaturas de revistas que contempla também a renda das editoras provenientes da venda de publicidade. Segundo Scott-Morton (2005) o nível de benefício de longo prazo que cada publicação oferece seria inversamente proporcional ao desconto oferecido pela editora na assinatura. Entretanto, não foram encontradas evidências empíricas deste comportamento ao serem consideradas outras características das publicações na análise econométrica em um mercado no qual os pontos de venda são muito mais disseminados. Em particular, o horizonte de assinatura não apresentou um papel relevante no grau de desconto da assinatura com relação ao preço de face. Ao contrário, uma análise exploratória feita por componentes principais, posteriormente confirmadas pelo modelo de regressão, sugere a existência de um processo multidimensional de discriminação de preços que envolve o desconto nas assinaturas e no preço de publicidade. Este é um trabalho em conjunto com Marcelo Resende (IE – UFRJ) e publicado na 38ª edição do Economics Bulletin.

###### 12/08

We discuss the effects of quenched disorder in a dilute Bose-Einstein condensate confined in a hard walls trap. Starting from the disordered Gross-Pitaevskii functional, we obtain a representation for the quenched free energy as a series of integer moments of the partition function. Positive and negative disorder-dependent effective coupling constants appear in the integer moments. Going beyond the mean-field approximation, we compute the static two-point correlation functions at first-order in the positive effective coupling constants. We obtain the combined contributions of effects due to boundary conditions and disorder in this weakly disordered condensate. The ground state renormalized density profile of the condensate is presented. We also discuss the appearance of metastable and true ground states for strong disorder, when the effective coupling constants become negative.

###### 01/07

O comportamento assintótico de classes de Passeios aleatórios não markovianos ou em ambiente aleatórios é um tema central em teoria das probabilidades e aplicações. Os passeios que iremos considerar neste seminário são tanto não markovianos quanto sua evolução pode ser descrita em função de um ambiente aleatório. Esses passeios são chamados de passeios aleatórios geradores de árvores e podem ser descritos da seguinte forma: (1) Consideramos um passeio aleatório em um grafo G com escolhas uniformes entre vizinhos próximos a cada transição; (2) após um certo número L (fixo) de passos um número de aleatório de vértices são criados e anexados à posição atual do passeio; (1) e (2) se repetem recursivamente. Estudaremos o comportamento assintótico desses passeios em função do valor de L, estabelecendo propriedades como recorrência, transiência e balisticidade.

###### 24/06

Most constructions of the zero range process assume that the rate at which a particle leaves a site grows at most linearly with the number of particles present at that site. We provide a method to construct a zero range processes with super-linear rates on $mathbb{Z}^d $ when either the initial distribution is translation invariant or d=1 and only nearest neighbor jumps are allowed.

###### 03/06

Inspired by Kalikow-type decompositions, we introduce a new stochastic model describing a network of interacting neurons. For such class we establish oracle inequalities for Lasso methods and restricted eigenvalue properties for the associated Gram matrix with high probability. These results hold even if the network is only partially observed. The main argument rely on the fact that concentration inequalities can easily be derived whenever the transition probabilities of the underlying process admit a sparse space-time representation.

###### 27/05

We consider an anisotropic finite-range bond percolation model on $mathbb{Z}^2$. On each horizontal layer $H_i={(x,i)colon x in mathbb{Z}}$ we have edges $langle (x,i),(y,i)rangle$ for $1 le |x-y|le N$. There are also vertical edges connecting two nearest neighbor vertices on distinct lines $langle (x,i),(x,i+1)rangle$ for $x,i in mathbb{Z}$. On this graph we consider the following anisotropic independent percolation model: horizontal edges are open with probability $1/(2N)$, while vertical edges are open with probability $epsilon$ to be suitably tuned as $N$ grows to infinity. The main result tells that if $epsilon = kappa N^{-frac25}$, then we see a phase transition in $kappa$: there exist positive and finite constants $C_1, C_2$ so that there is no percolation if $kappa

###### 13/05

A well-known conjecture states that a random symmetric n-by-n matrix with entries in {-1,1} is singular with probability 2^{-n+o(1)}. In this talk we will show that the probability of this event is at most 2^{-cn^(1/2)}, improving the best known bound 2^{-cn^(1/4)}, which was obtained recently by Ferber and Jain. The main new ingredient is an inverse Littlewood–Offord theorem in Z_p^n that applies under very mild conditions, whose statement is inspired by the method of hypergraph containers. This is a joint work with Marcelo Campos, Robert Morris and Natasha Morrison.

###### 01/04

Bootstrap percolation is a monotone version of the Glauber dynamics of the Ising model of ferromagnetism. In this talk we will consider anisotropic bootstrap models, which are three-dimensional analogues of a family of (two-dimensional) processes studied by Duminil-Copin, van Enter and Hulshof. In these models the underlying graph $G$ has vertex set $[L]^3$, and the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $e_i$-direction for each $i in {1,2,3}$, where $a_1le a_2le a_3$. Given an initial configuration in ${0,1}^{V(G)}$, the system evolves in discrete time in the following way: the state of a vertex $v$ changes from $0$ to $1$ when it has at least $r$ neighbours in state $1$. The initial state is usually chosen to be the product of Bernoulli measures with density $p$, and the main question is to determine the so-called {it critical length for percolation} $L_c(p)$, for small values of $p$.

It turns out that $L_c(p)$ is polynomial if $r le a_3$, exponential if $a_3 < r le a_2 + a_3$, doubly exponential if $a_2 + a_3 < r le a_1 + a_2 + a_3$, and infinite if $r > a_1 + a_2 + a_3$. In this talk we will focus on the case $r = a_3 + 1$, and show how to determine $log L_c(p)$ up to a constant factor. The main new tool, which we call the {it beams process}, allows one to reduce the problem to proving an exponential decay property for a certain two-dimensional model whose behaviour resembles site percolation.

###### 18/03

We consider the following dynamic. We start with a fix number of la-beled, i.i.d. Markov chains over a finite state space, let the time pass, and when two chains meet, they behave as one chain. This dynamic induces a process in the set of partitions of the first natural numbers. We are interested in the asymptotic behavior of this process. On the late eighties J. T. Cox obtained some limit theo rems for coalescing random walks on the discrete torus, when the Markov chains we considered before are simple random walks and we start with one random walk in each vertex of the torus. Since then the asymptotic behavior of the coa-lescence time, the first time all the chains meet, has been the subject of several papers. And the result of Cox has being extended in different ways. In this talk we describe some of these extensions, and use a martingale approach to prove, under certain conditions, the convergence of the process in the set of partitions of the first natural numbers, we described before, to the Kingman’s coalescent start-ing from a finite number of partitions.

###### 11/03

We consider a symmetric finite-range contact process on Z with two types of particles (or infections), which propagate according to the same supercritical rate and die (or heal) at rate 1. Particles of type 1 can occupy any site in (-infty,0]that is empty or occupied by a particle of type 2 and, analogously, particles of type 2 can occupy any site in [1,+infty) that is empty or occupied by a particle of type 1. We consider the model restricted to a finite interval [-N+1,N] (on the integers). If the initial configuration is (-N,0] fully occupied by type 1 particles and [1,N) fully occupied by type 2 particles, we prove that this dynamic presents two metastable states: one with the two species and the other one with the family that survives the competition.