Variance of the range of a random walk

T/sub n/, the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 through n, is obtained from its relation to the dual first occupancy probability F/sub i//sub j/(x,x'), and the latter quantity is obtained from a recursion with the first occupancy probability F/sub k/(x''). The variance V/sub n/ of the number of distinct sites occupied is calculated directly from T/sub n/; the procedure is illustrated by the calculation of V/sub n/ (4096≥n) and the derivation of asymptotic expansions for V/sub n/ for a particular random walk in dimensions 1 through 3