Random walks on lattices with points of two colors. II. Some rigorous inequalities for symmetric random walks

We continue our investigation of a model of random walks on lattices with two kinds of points, black and white. The colors of the points are stochastic variables with a translation-invariant, but otherwise arbitrary, joint probability distribution. The steps of the random walk are independent of the colors. We are interested in the stochastic properties of the sequence of consecutive colors encountered by the walker. In this paper we first summarize and extend our earlier general results. Then, under the restriction that the random walk be symmetric, we derive a set of rigorous inequalities for the average length of the subwalk from the starting point to a first black point and of the subwalks between black points visited in succession. A remarkable difference in behavior is found between subwalks following an odd-numbered and subwalks following an evennumbered visit to a black point. The results can be applied to a trapping problem by identifying the black points with imperfect traps.