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By Daniel J. Velleman

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Unpredictable paths and percolation, Ann. Probab. 26 (1998) 1198–1211. 1214/aop/ 1022855749 4. P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984. ¨ 5. F. Eggenberger and G. P´olya, Uber die statistik vorketter vorg¨ange, Zeit. Angew. Math. Mech. 3 (1923) 279–289. 19230030407 6. O. H¨aggstr¨om and E. Mossel, Nearest-neighbor walks with low predictability profile and percolation in 2 + dimensions, Ann.

Fix a ∈ V . Let Z n = {v : d(a, v) ≥ n}, and create a finite graph Gn by identifying all vertices in Z n with a new vertex z n , and removing all edges with both vertices in Z n . Let In be the unit current flow on Gn from a to z n . If Nn (x, y) is the net number of traversals of the oriented edge (x, y) in Gn before hitting z n , then In (x, y) = Ea Nn (x, y). With a minor abuse of notation, Nn (x, y) is also the net number of traversals of the oriented edge (x, y) in G before hitting Z n ; with this interpretation, Nn (x, y) ↑ N (x, y), where N (x, y) is the net number of traversals of (x, y) for the random walk in G started at a and run for infinite time.

A. Nash-Williams, Random walk and electric currents in networks, Proc. Cambridge Philos. Soc. 55 (1959) 181–194. 1017/S0305004100033879 14. J. R. Norris, Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2, Cambridge University Press, Cambridge, 1998; reprint of 1997 original. 15. R. Pemantle, A survey of random processes with reinforcement, Probab. Surv. 4 (2007) 1–79. 1214/07-PS094 16. J. Pitman, Combinatorial Stochastic Processes, Lecture Notes in Mathematics, vol.