{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,18]],"date-time":"2025-11-18T09:18:19Z","timestamp":1763457499297,"version":"build-2065373602"},"reference-count":0,"publisher":"European Mathematical Society - EMS - Publishing House GmbH","issue":"10","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Eur. Math. Soc."],"published-print":{"date-parts":[[2015,10,29]]},"abstract":"<jats:p>\n                    In this paper we establish a decoupling feature of the random interlacement process\n                    <jats:inline-formula>\n                      <jats:tex-math>\\mathcal{I}^u \\subset \\mathbb Z^d<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    at level\n                    <jats:inline-formula>\n                      <jats:tex-math>u<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:tex-math>d \\geq 3<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . Roughly speaking, we show that observations of\n                    <jats:inline-formula>\n                      <jats:tex-math>\\mathcal{I}^u<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    restricted to two disjoint subsets\n                    <jats:inline-formula>\n                      <jats:tex-math>A_1<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>A_2<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    of\n                    <jats:inline-formula>\n                      <jats:tex-math>\\mathbb Z^d<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    are approximately independent, once we add a sprinkling to the process\n                    <jats:inline-formula>\n                      <jats:tex-math>\\mathcal{I}^u<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    by slightly increasing the parameter\n                    <jats:inline-formula>\n                      <jats:tex-math>u<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . Our results differ from previous ones in that we allow the mutual distance between the sets\n                    <jats:inline-formula>\n                      <jats:tex-math>A_1<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>A_2<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    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\n                    <jats:inline-formula>\n                      <jats:tex-math>u_{**}<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , the probability of having long paths that avoid\n                    <jats:inline-formula>\n                      <jats:tex-math>\\mathcal{I}^u<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    is exponentially small, with logarithmic corrections for\n                    <jats:inline-formula>\n                      <jats:tex-math>d=3<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>\n                  <jats:p>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 \u201csmoothened\u201d 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.<\/jats:p>","DOI":"10.4171\/jems\/565","type":"journal-article","created":{"date-parts":[[2015,10,29]],"date-time":"2015-10-29T07:10:28Z","timestamp":1446102628000},"page":"2545-2593","source":"Crossref","is-referenced-by-count":52,"title":["Soft local times and decoupling of random interlacements"],"prefix":"10.4171","volume":"17","author":[{"given":"Serguei","family":"Popov","sequence":"first","affiliation":[{"name":"University of Campinas, Brazil"}]},{"given":"Augusto","family":"Teixeira","sequence":"additional","affiliation":[{"name":"IMPA, Rio de Janeiro, Brazil"}]}],"member":"2673","container-title":["Journal of the European Mathematical Society"],"original-title":[],"link":[{"URL":"http:\/\/www.ems-ph.org\/fulltext\/10.4171\/JEMS\/565","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T13:58:44Z","timestamp":1762869524000},"score":1,"resource":{"primary":{"URL":"https:\/\/ems.press\/doi\/10.4171\/jems\/565"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,29]]},"references-count":0,"journal-issue":{"issue":"10"},"URL":"https:\/\/doi.org\/10.4171\/jems\/565","relation":{},"ISSN":["1435-9855","1435-9863"],"issn-type":[{"type":"print","value":"1435-9855"},{"type":"electronic","value":"1435-9863"}],"subject":[],"published":{"date-parts":[[2015,10,29]]}}}