Random Walks with Spontaneous Emission

the rate of one step every T2 seconds. We assume, as usual, that two kinds of sites exist in the lattice, "6ordinary" or "good" ones and a fraction q of "'absorbhig" or "bad" ones, the latter having the property of absorbing the particle if it chances upon one of them, thus terminating the walk. In addition we assume the existence of another possible way of terminating the walk, namely a spontaneous emission probability rT1 per second for the particle to be emitted from the lattice (i.e., jump out of the lattice). We wish to compute the probability P(T1, T2, q) of emission (rather than absorption) taking place. The problem thus concerns a symmetrical random walk on an n-dimensional lattice, n = 1, 2, 3, with absorbing points, and, in addition, emission. Alternatively, it may be formulated as an unsymmetrical random walk on a lattice of n + 1 dimensions, emission corresponding to a step into the (n + 1)st dimension. The formulation used in the following developments will be the first one.