Soft local times and decoupling of random interlacements

Popov, Serguei, and Teixeira, Augusto. "Soft local times and decoupling of random interlacements." Journal of the European Mathematical Society 017.10 (2015): 2545-2593. <http://eudml.org/doc/277751>.

@article{Popov2015,
abstract = {In this paper we establish a decoupling feature of the random interlacement process $\mathcal \{I\}^u \subset \mathbb \{Z\}^d$ at level $u$, $d \ge 3$. Roughly speaking, we show that observations of $\mathcal \{I\}^u$ restricted to two disjoint subsets $A_1$ and $A_2$ of $\mathbb \{Z\}^d$ are approximately independent, once we add a sprinkling to the process $\mathcal \{I\}^u$ 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 $\mathcal \{I\}^u$ 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.},
author = {Popov, Serguei, Teixeira, Augusto},
journal = {Journal of the European Mathematical Society},
keywords = {random interlacements; stochastic domination; soft local time; connectivity decay; smoothening of discrete sets; random interlacements; decoupling; stochastic domination; Markov chains; soft local times; connectivity decay; smoothening of discrete sets},
language = {eng},
number = {10},
pages = {2545-2593},
publisher = {European Mathematical Society Publishing House},
title = {Soft local times and decoupling of random interlacements},
url = {http://eudml.org/doc/277751},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Popov, Serguei
AU - Teixeira, Augusto
TI - Soft local times and decoupling of random interlacements
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 10
SP - 2545
EP - 2593
AB - In this paper we establish a decoupling feature of the random interlacement process $\mathcal {I}^u \subset \mathbb {Z}^d$ at level $u$, $d \ge 3$. Roughly speaking, we show that observations of $\mathcal {I}^u$ restricted to two disjoint subsets $A_1$ and $A_2$ of $\mathbb {Z}^d$ are approximately independent, once we add a sprinkling to the process $\mathcal {I}^u$ 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 $\mathcal {I}^u$ 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.
LA - eng
KW - random interlacements; stochastic domination; soft local time; connectivity decay; smoothening of discrete sets; random interlacements; decoupling; stochastic domination; Markov chains; soft local times; connectivity decay; smoothening of discrete sets
UR - http://eudml.org/doc/277751
ER -