The average number of distinct sites visited by a one-dimensional random walker and its application to isotope exchange in polypeptides

Abstract

The average number of distinct sites visited by a random walker moving with arbitrary transition probability on a one-dimensional lattice is calculated. Asymptotic forms of this quantity for both asymmetric and symmetric random walks are determined, and an exact solution for the latter case is also given for any number of steps. The average number of sites visited is then analyzed for intermediate numbers of steps by introducing an exponent. This approach is applied to explain the results of isotope exchange experiments in polypeptides, and applications of asymmetric random walks to other biological problems are briefly discussed.