Optimal path and directed percolation

An efficient transfer matrix technique is introduced to study directed optimal paths in two and three dimensions. The roughness exponent zeta is 0.6325 +/- 0.0007 for the two-dimensional case and zeta = 0.555 +/- 0.008 for the three-dimensional one, in agreement with the recent conjecture zeta = v(perpendicular to)/v(parallel to), where v(perpendicular to) and v(parallel to) are the correlation length exponents of directed percolation. Exactly solvable examples are also analyzed