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 [Commun. Math. Phys. 343 (2016) 129–164]. 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.
"The vacant set of two-dimensional critical random interlacement is infinite." Ann. Probab. 45 (6B) 4752 - 4785, November 2017. https://doi.org/10.1214/17-AOP1177