Random walks on lattices with points of two colours. I

This paper is concerned with random walks on lattices with two kinds of points, black and white. The colours of the points are random variables with a translation invariant, but otherwise arbitrary, joint probability distribution. The steps of the walk are independent of the colours. We study the stochastic properties of the length of the subwalk from the starting point to a first black point and of subwalks between points visited in succession, and establish a number of exact relations. These relations can be applied to a trapping problem by identifying the black points with imperfect traps. An example is discussed.