Directed compact percolation: cluster size and hyperscaling

Exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice. The corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained. The work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice. A duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve. The moments of both distributions have critical exponents which satisfy a c o t ” gap hypothesis with gap exponents U,-, = 2 and A = 3 corresponding to the scaling length and scaling size respectively. The usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by A = + Du, where U, is the exponent corresponding to the cluster width and D is the number of transverse dimensions ( = 1 for the square lattice). 1.