Random Walks Strictly Confined to a Subspace

In this paper, we show that for (i) simple random walks and (ii) random walks with exclusion of immediate reversals it is possible to partition the set of walks into different sub-lattices according to the number of axis directions in which the steps are made. We also show that for (i) and (ii) the cardinality of the set of walks confined to such a sub-lattice are very similar in structure to Stirling numbers of the second kind. We also show that these cardinalities are unimodal for (i) and (ii). Finally, we estimate the probability of and the conditional expectation of the end-to-end distance squared for a walk is that strictly confined to a ‘' dimensional sub-space in both cases