On the Shapiro problem of random walks with absorbing points

  • Authors:

  • Chizhong Zhou

  • Affiliations:

  • Department of Computer and Information Engineering, Yueyang Normal University, Yueyang, Hunan 414000, P. R. China

  • Venue:
  • Analysis, combinatorics and computing
  • Year:
  • 2002

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Abstract

In this paper, we study random walks on the lattice Z2 starting from (0, 0) and with steps (1, 0), (1, 1) and (0, 1), each with weight A, B, C, respectively. Divide all the lattice points on line y = x for x 0 into k equivalence classes by modulo k, and select out one class or more classes from the k classes to be the set of absorbing points. L. Shapiro has enumerated the sum of the weights of all safe paths for the case k = 2 and given applications to probability in [7]. Here we solve the problem for any k and for more complicated set of absorbing points, and develop its applications to probability.