The relevance of Polya's random-walk problem for the single-species reaction-diffusion system

The diffusion-limited reactions A + A --> A and A + A --> 0 in dimension d > 2 are reconsidered from the point of view of the random-walk theory. It is pointed out that Polya's theorem on the returning probability of a random walker to the origin, which would imply a probability less than one for the meeting of two typical particles, would predict the possibility of a state in which the reaction seems to have spontaneously ceased, in contradiction with the very well known asymptotic N(t) similar to t(-1) for the particles population of these reactions. In fact, a given condition is presented, in which the relative particle number N(t)/N(0) decays to a non-vanishing constant. The condition is that the initial distribution of particles in the d-dimensional space has a dimension gamma, such that 0 < gamma < d - 2.