Ergodic properties of color records

In a series of papers culminating in the dissertation of the second author under the direction of P.W Kasteleyn, several rigorous results have been obtained concerning the color sequence seen by a random walker on a stochastically black-white colored lattice, and this under the sole assumption that the coloring is stationary, ergodic and independent of the walking. In this article we investigate the ergodic properties of the color record process and their implications for the behavior of the sequence nk, k ⩾0, of successive times needed for the walker to go from the kth to the (k + 1)st black point. As an example of the type of results obtained, we show that if a certain “induced” color record process is strongly mixing in the ergodic theoretic sense, then the average 〈nk〉 tends to the inverse of the density of black points as k tends to infinity, and, similarly, that the distribution of nk tends to a limit that can be explicitly calculated in terms of the distribution of n0. For a certain class of stochastic colorings and random walks we are able to verify that the induced color record process is indeed strongly mixing. Our main goal, however, is to clarify the nature of the ergodic properties and thereby provide a tool for future use in the study of color records.