RANDOM WALKS ON LATTICES .3. CALCULATION OF FIRST-PASSAGE TIMES WITH APPLICATION TO EXCITON TRAPPING ON PHOTOSYNTHETIC UNITS

The following statistical problem arises in the theory of exciton trapping in photosynthetic units: Given an infinite periodic lattice of unit cells, each containing N points of which (N-1) are chlorophyll molecules and one is a trap; if an exciton is produced with equal probability at any nontrapping point, how many steps on the average are required before the exciton reaches a trapping center for the first time? It is shown that, when steps can be taken to near-neighbor lattice points only, as N→∞, our required number of steps is <n>∼{N2/6, linear chain, π-1NlogN, square lattice, 1.516N, single cubic lattice. The correction terms for medium and relatively small N are obtained for a number of lattices.