The vacant set of two-dimensional critical random interlacement is infinite

Abstract

For the model of two-dimensional random interlacements in the critical regime (i.e., $\alpha =1$), we prove that the vacant set is a.s.\ infinite, thus solving an open problem from arXiv:1502.03470. Also, we prove that the entrance measure of simple random walk on annular domains has certain regularity properties; this result is useful when dealing with soft local times for excursion processes.

38 pages, 3 figures; to appear in The Annals of Probability

References in corpus (4)

• Late points for random walks in two dimensions
• Two-dimensional random interlacements and late points for random walks
• The subleading order of two dimensional cover times
• An improved decoupling inequality for random interlacements

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• Second order term of cover time for planar simple random walk
• On pinned fields, interlacements, and random walk on $(\mathbb{Z}/N\mathbb{Z}{)}^2$
• Percolation for two-dimensional excursion clouds and the discrete Gaussian free field