On the Dependence Structure in Random Interlacements and the Meeting Time of Random Walks in Random Environments

• Grant number: 14/14323-9
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Start date: October 01, 2014
• End date: September 30, 2015
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Probability
• Principal Investigator: Serguei Popov
• Grantee: Diego Fernando de Bernardini
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

In this research project we discuss some issues which are considered relevant in the context of two well-known stochastic models: the random interlacements model and the random walk in random environments model. In the context of random interlacements, at first we propose to tackle two problems. The first problem is related to the characterization of the covariance between two convex events in this process, in terms of its decay with respect to the distance between the disjoint sets on which such events are supported. The second problem, on the other hand, refers to the investigation of the disconnection and connectivity decay on the vacant set of random interlacements, and in this case specifically we intend to improve two particular results recently published in the literature. As the main difficulty in the study of random interlacements lies in the fact that such model is a dependent percolation model, which exhibits, in particular, long-range dependence, we hope the results we intend to obtain will help to understand more clearly certain properties of the process. On the other hand, in the context of the one-dimensional random walk in random environment, we intend to complement the advances obtained in the literature concerning the investigation of the moments of the first meeting time of one pair of walks among N independent walks, in the usual Sinai's model. Additionally, we intend to study such moments when this model is subjected to a specific external potential.

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

DE BERNARDINI, DIEGO F.; POPOV, SERGUEI. Russo's Formula for Random Interlacements. Journal of Statistical Physics, v. 160, n. 2, p. 321-335, JUL 2015. (14/14323-9, 09/52379-8)

DE BERNARDINI, DIEGO F.; GALLESCO, CHRISTOPHE; POPOV, SERGUEI. An Improved Decoupling Inequality for Random Interlacements. Journal of Statistical Physics, v. 177, n. 6, p. 1216-1239, DEC 2019. (17/02022-2, 14/14323-9, 17/10555-0, 17/19876-4)