| Markov chain intersections and the loop-erased walk (2008) | |||||||||||||||
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| Abstract. Let X and Y be independent transient Markov chains on the same state space that have the same transition probabilities. Let L denote the “loop-erased path ” obtained from the path of X by erasing cycles when they are created. We prove that if the paths of X and Y have infinitely many intersections a.s., then L and Y also have infinitely many intersections a.s. Résumé. Soit X et Y deux chaînes de Markov indépendantes et transientes à même espace d’états et aux mêmes probabilités de transition. Soit L le “chemin aux boucles effacées ” obtenu du chemin de X en effaçant les cycles aux moments de création. Nous démontrons que si les chemins de X et Y ont une infinité d’intersections p.s., alors L et Y ont aussi une infinité d’intersections p.s. §1. Introduction. Erdős and Taylor [3] proved that two independent simple random walk paths in Z d intersect infinitely often if d ≤ 4, but not if d> 4. Lawler [9] proved that for d = 3, 4, there are still infinitely many intersections even if one of the paths is replaced by its | |||||||||||||||
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