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 $\zeta =0.555\mathrm{}\pm 0.008$ for the three-dimensional one, in agreement with the recent conjecture $\zeta =\frac{{\nu }_{\perp }}{{\nu }_{\parallel }}$, where ${\nu }_{\perp }$ and ${\nu }_{\parallel }$ are the correlation length exponents of directed percolation. Exactly solvable examples are also analyzed.

• Received 8 November 1995

DOI:https://doi.org/10.1103/PhysRevE.53.R2029