Probability Seminar "Stein’s method and asymptotic independence" November 6, from 3:30 p.m. to 4:30 p.m. (Rio de Janeiro local time)
Local: Google Meet
Speaker: Ciprian Tudor (Université de Lille)
Abstract: If Y is a random vector in R^d we denote by P_Y its probability distribution. Consider a random variable X and a d-dimensional random vector Y. We develop a multidimensional variant of the Stein-Malliavin calculus which allows to measure the Wasserstein distance between the law P_(X, Y) and the probability distribution P_Z x P_Y, where Z is a Gaussian random variable. That is, we give estimates, in terms of the Malliavin operators, for the distance between the law of the random vector (X, Y) and the law of the vector (Z,Y), where Z is Gaussian and independent of Y.
Then we focus on the particular case of random vectors in Wiener chaos and we give an asymptotic version of this result. In this situation, this variant of the Stein-Malliavin calculus has strong and unexpected consequences.
More complete information about the seminars can be found at DME.
Title: "Properties of the gradient squared of the Gaussian free field"
Speaker: Sergio I. López (Universidad Nacional Autónoma de México)
Date: 30/10/2023
Time: 3:30 p.m. to 4:30 p.m. (Rio de Janeiro local time)
Local: Online
This meeting will take place on Google Meet, through the link HERE.
Abstract: In this talk we study the scaling limit of a random field which is a non-linear transformation of the gradient Gaussian free field. More precisely, our object of interest is the recentered square of the norm of the gradient Gaussian free field at every point of the square lattice. Surprisingly, in dimension 2 this field bears a very close connection to the height-one field of the Abelian sandpile model studied in Dürre (2009). In fact, with different methods we are able to obtain the same scaling limits of the height-one field: on the one hand, we show that the limiting cumulants are identical (up to a sign change) with the same conformally covariant property, and on the other that the same central limit theorem holds when we view the interface as a random distribution. We generalize these results to higher dimensions as well.
Joint work with Rajat Subhra Hazra (Leiden), Alan Rapoport (Utrecht) and Wioletta Ruszel (Utrecht)
More complete information about the seminars can be found at HERE
Sincerely,
Organizers: Giulio Iacobelli e Maria Eulalia Vares
Titulo: Interacting Edge-Reinforced Random Walks
Palestrante: Guilherme Reis (Technical University of Munich)
Data: 04/09/2023
Horário: 3:30 p.m. to 4:30 p.m.
Sala: C116 - Bloco C - CT – Instituto de Matemática – UFRJ. There will be no transmission online.
Resumo: Recall that in the simple Random Walk (RW) on Z the walker, starting at 0, just jumps either to the right or to the left with the same probability. It is a classical result that the simple RW on Z is recurrent. In the Edge-Reinforced Random Walk (ERRW) the walker keeps track of the edges already visited and gives extra bias to the edges mostly visited. We would expect that the behavior of the ERRW depends on the strength of the extra bias we decide to give to the edges. The ERRW is a non-markovian process introduced by Diaconis and the first results about it goes back to Davis in 1990. Davis showed, under some assumptions, that the ERRW on Z is either recurrent or it localizes in a single edge with probability 1. What would happen if instead of a single ERRW we consider two or more walkers reinforcing the edges of Z? In an ongoing project, together with Nina Gantert (TUM) and Fabian Michel (TUM), we plan to answer the above question.
More complete information about the seminars can be found at clicking HERE.
Sincerely,
Organizers: Giulio Iacobelli e Maria Eulalia Vares
Title: Targeted cutting of random recursive trees.
Speaker: Sergio I. López (Universidad Nacional Autónoma de México)
Date: 16/10/2023
Horário: 3:30 p.m. to 4:30 p.m. (Rio de Janeiro local time)
Local: C116 - Bloco C - CT – Instituto de Matemática – UFRJ. There will be no transmission online.
Abstract: Increasing trees are rooted trees, where each vertex has a unique label and the labels along paths away from the root are in increasing order. A Random recursive tree on ? vertices (abbreviated as RRTs) is a tree chosen uniformly at random from the set of increasing trees with vertex labels {1,…,?}. The idea of cutting random recursive trees was introduced by Meir and Moon in 1974. They studied the following procedure: Start with a random recursive tree on ? vertices. Choose an edge at random and remove it, along with the cut subtree that does not contain the root. Repeat until the remaining tree consists only of the root; at which point, we say that the tree has been deleted.
Let X be the number of edge removals needed to delete a RRT with ? vertices. The random variable X has been thoroughly studied and analogous variables under distinct models of random trees have been analyzed; in particular, X grows asymptotically as ? ln(?). In this talk we propose and study a method for cutting down a random recursive tree that focuses on its largest degree vertices. Enumerate the vertices of a random recursive tree of size ? according to a decreasing order of their degrees. The targeted, vertex-cutting process is performed by sequentially removing vertices according to that order and keeping only the subtree containing the root after each removal. The algorithm ends when the root is picked to be removed.
Joint work with Laura Eslava and Marco L. Ortiz.
More complete information about the seminars can be found at HERE
Sincerely,
Organizers: Giulio Iacobelli e Maria Eulalia Vares
Títle: Minimal distance between random orbits
Palestrante: Manuel Stadlbauer (IM-UFRJ)
Data: 21/08/2023
Horário: 15:30 às 16:30h
Local: C116 - Bloco C - CT – Instituto de Matemática – UFRJ.
Resumo: It is known for expanding dynamical systems and finite state Markov chains that the asymptotic behaviour of the minimal distance between two orbits up to time n is given by its correlation dimension.
In this talk, we will discuss this problem in a randomized setting with not necessarily expanding fibres. If the fibres and the basis of the random system
under consideration are sufficiently mixing, then a similar but more complex result holds: there are two relevant dimensions and, depending on the stochastic process in the
basis, either one or the other is dominant. In particular, there is a phase transition, which is unknown in the framework of a classical dynamical system.
Informações mais completas sobre os seminários estão disponíveis AQUI.
Sincerely,
Organizers: Giulio Iacobelli e Maria Eulalia Vares