# Connectivity Bounds for the Vacant Set of Random Interlacements - Mathematics > Probability

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Abstract: The model of random interlacements on Z^d, d bigger or equal to 3, wasrecently introduced in arXiv:0704.2560. A non-negative parameter u parametrizesthe density of random interlacements on Z^d. In the present note we investigatethe connectivity properties of the vacant set left by random interlacements atlevel u, in the non-percolative regime, where u is bigger than thenon-degenerate critical parameter for percolation of the vacant set, seearXiv:0704.2560, arXiv:0808.3344. We prove a stretched exponential decay of theconnectivity function for the vacant set at level u, when u is bigger than another critical parameter. It is presently an open problem whether these twocritical parameters actually coincide.

Author: ** Vladas Sidoravicius, Alain-Sol Sznitman**

Source: https://arxiv.org/