Soft local times and decoupling of random interlacements

Author(s)

Popov, Serguei ^[1] ; Teixeira, Augusto ^[2]

Affiliation

^[1] Univ Campinas UNICAMP, Inst Math Stat & Sci Computat, Dept Stat, BR-13083859 Campinas, SP - Brazil

^[2] Inst Nacl Matemat Pura & Aplicada IMPA, BR-22460320 Rio De Janeiro, RJ - Brazil

• Document type: Journal article
• Source: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY; v. 17, n. 10, p. 2545-2593, 2015.
• Web of Science Citations: 15

Abstract

In this paper we establish a decoupling feature of the random interlacement process I-u subset of Z(d) at level u, d >= 3. Roughly speaking, we show that observations of I-u restricted to two disjoint subsets A(1) and A(2) of Z(d) approximately independent, once we add a sprinkling to the process I-u by slightly increasing the parameter u. Our results differ from previous ones in that we allow the mutual distance between the sets A(1) and A(2) to be much smaller than their diameters. We then provide an important application of this decoupling for which such flexibility is crucial. More precisely, we prove that, above a certain critical threshold u({*}{*}), the probability of having long paths that avoid I-u is exponentially small, with logarithmic corrections for d = 3. To obtain the above decoupling, we first develop a general method for comparing the trace left by two Markov chains on the same state space. This method is based on what we call the soft local time of a chain. In another crucial step towards our main result, we also prove that any discrete set can be ``smoothened{''} into a slightly enlarged discrete set, for which its equilibrium measure behaves in a regular way. Both these auxiliary results are interesting in themselves and are presented independently of the rest of the paper. (AU)

• FAPESP's process: 09/52379-8 - Stochastic modeling of interacting systems
• Grantee: Fabio Prates Machado
• Support Opportunities: Research Projects - Thematic Grants