Escape probabilities for slowly recurrent sets

Summary A setAZ ^d (d>-3) is defined to be slowly recurrent for simple random walk if it is recurrent but the probability of enteringA{z:n<|z|<-2n} tends to zero asn. A method is given to estimate escape probabilities for such sets, i.e., the probability of leaving the ball of radiusn without entering the set. The methods are applied to two examples. First, half-lines and finite unions of half-lines inZ ³ are considered. The second example is a random walk path in four dimensions. In the latter case it is proved that the probability that two random walk paths reach the ball of radiusn without intersecting is asymptotic toc(lnn)^–1/2, improving a result of the author.Research partially supported by the National Science Foundation