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Soft local times and decoupling of random interlacements

Serguei Popov;  Augusto Teixeira — 2015

Journal of the European Mathematical Society

In this paper we establish a decoupling feature of the random interlacement process ℐ u ⊂ ℤ d at level u , d ≥ 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...

Giant vacant component left by a random walk in a random d-regular graph

Jiří Černý;  Augusto Teixeira;  David Windisch — 2011

Annales de l'I.H.P. Probabilités et statistiques

We study the trajectory of a simple random walk on a -regular graph with ≥ 3 and locally tree-like structure as the number of vertices grows. Examples of such graphs include random -regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time , where > 0 is a fixed positive parameter. We show that this so-called set exhibits a phase transition in in the following sense: there exists an explicitly...

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