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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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
random-walk loop soup; loop percolation
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