Soft local times and decoupling of random interlacements

2012/12/31 by Serguei Popov, Augusto Teixeira · 4 citations
Decision Sciences · Mathematics · #Combinatorics #Computer science #Decoupling (probability) #Discrete mathematics #Disjoint sets #Logarithm #Markov Chains and Monte Carlo Methods #Markov chain #Markov process #Mathematical analysis #Mathematics #Measure (data warehouse) #Probability and Risk Models #Statistics #Stochastic processes and statistical mechanics #math.PR #msc:60G50 #msc:60K35 #msc:82C41

paper · pdf · doi:10.4171/jems/565

published as Journal of the European Mathematical Society, Volume 17, Issue 10, 2015, pp. 2545-2593 · 10 figures

arxiv created 2015/07/28 · arxiv updated 2015/09/29 · openalex publication_date 2015/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper we establish a decoupling feature of the random interlacement process I^u ⊂ \mathbb Z^d at level u , d ≥ 3 . Roughly speaking, we show that observations of I^u restricted to two disjoint subsets A₁ and A₂ of \mathbb Z^d are 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 A₁ and A₂ 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 in 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 from the rest of the paper.

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• Decoupling inequalities for the Ginzburg-Landau ∇ φ models
• Excess deviations for points disconnected by random interlacements
• Non triviality of the percolation threshold and Gumbel fluctuations for Branching Interlacements

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