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Loop percolation on discrete half-plane. (English) Zbl 1338.60235
Summary: We consider the random walk loop soup on the discrete half-plane \(\mathbb{Z} \times \mathbb{N} ^{\ast }\) and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to \(\frac{1} {2}\). The absence of percolation at intensity \(\frac{1} {2}\) was shown in a previous work. We also show that in the supercritical regime, one can keep only the loops up to some large enough upper bound on the diameter and still have percolation.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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