Soft local times and decoupling of random interlacements

Abstract

In this paper we establish a decoupling feature of the random interlacement process I^u in Z^d, at level u, for d \geq 3. Roughly speaking, we show that observations of I^u restricted to two disjoint subsets A_1 and A_2 of 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 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 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.

10 figures

References in corpus (6)

• Decoupling inequalities and interlacement percolation on G x Z
• On the uniqueness of the infinite cluster of the vacant set of random interlacements
• On the fragmentation of a torus by random walk
• On the internal distance in the interlacement set
• Giant vacant component left by a random walk in a random d-regular graph
• The effect of small quenched noise on connectivity properties of random interlacements

Cited by in corpus (41)

• On chemical distances and shape theorems in percolation models with long-range correlations
• On decoupling inequalities and percolation of excursion sets of the Gaussian free field
• Equality of critical parameters for percolation of Gaussian free field level-sets
• On large deviations for the cover time of two-dimensional torus
• Two-dimensional random interlacements and late points for random walks
• The sign clusters of the massless Gaussian free field percolate on ${\mathbb{Z}}^d$, $d⩾3$ (and more)
• On the radius of Gaussian free field excursion clusters
• The vacant set of two-dimensional critical random interlacement is infinite
• A lower bound for disconnection by random interlacements
• On macroscopic holes in some supercritical strongly dependent percolation models
• Conditional decoupling of random interlacements
• Geometry of Gaussian free field sign clusters and random interlacements
• Biased random walk on the interlacement set
• On uniform closeness of local times of Markov chains and i.i.d. sequences
• Interface motion in random media
• A short proof of the phase transition for the vacant set of random interlacements
• A lower bound for disconnection by simple random walk
• Excess deviations for points disconnected by random interlacements
• On bulk deviations for the local behavior of random interlacements
• Disconnection and entropic repulsion for the harmonic crystal with random conductances
• Russo's formula for random interlacements
• Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
• Two-dimensional Brownian random interlacements
• Percolation for the Gaussian free field on the cable system: counterexamples
• How can a clairvoyant particle escape the exclusion process?
• Phase transition for level-set percolation of the membrane model in dimensions $d\ge 5$
• Percolative properties of Brownian interlacements and its vacant set
• Spread of an infection on the zero range process
• Random walks in random conductances: decoupling and spread of infection
• Random interlacement is a factor of i.i.d
• First passage percolation with long-range correlations and applications to random Schrödinger operators
• Phase transition in loop percolation
• An improved decoupling inequality for random interlacements
• Branching interlacements and tree-indexed random walks in tori
• Sharp threshold for two-dimensional majority dynamics percolation
• First passage percolation, local uniqueness for interlacements and capacity of random walk
• Local survival of spread of infection among biased random walks
• On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements
• Disconnection and level-set percolation for the Gaussian free field
• On the percolative properties of the intersection of two independent interlacements
• On the threshold of spread-out voter model percolation