| COMMUNICATIONS in PROBABILITY LOOP-ERASED WALKS INTERSECT INFINITELY OFTEN IN FOUR DIMENSIONS (2008) | |||||||||||||
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| In this short note we show that the paths two independent loop-erased random walks in four dimensions intersect infinitely often. We actually prove the stronger result that the cut-points of the two walks intersect infinitely often. Let S(t) be a transient Markov chain with integer time t on a countable state space. Associated to S, is the loop-erased process ˆ S obtained by erasing loops in chronological order defined as follows. Let s0 =sup{t: S(t)=S(0)}, and for n>0, sn =sup{t: S(t)=S(sn−1 +1)}. Then the loop-erased process ˆS(n) is defined by ˆS(n)=S(sn). This is well-defined with probability one by transience. (In many cases the process is welldefined for recurrent chains with a slightly modified definition, but we are only interested in transient chains here.) Note that the path of the loop-erased process is contained in the path of the original process. The loop-erased process was first studied when S is simple random walk on the integer lattice (see [2] and references therein), but recent results relating looperased processes to uniform spanning trees has caused an interest in the loop-erased process for arbitrary Markov chains. Lyons, Peres, and Schramm [6] have recently shown that if S 1,S 2 are two independent realizations of the Markov chain starting at the same point then the probability that S 1 [0, ∞) ∩ | |||||||||||||
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