Domany-Kinzel Model of Directed Percolation: Formulation as a Random-Walk Problem and Some Exact Results

It is shown that the directed percolation on certain two-dimensional lattices, in which the occupation probability is unity along one spatial direction, is related to a random-walk problem, and is therefore exactly solvable. As an example, the case of the triangular lattice is solved. It is also shown that the square-lattice solution obtained previously by Domany and Kinzel can be derived using Minkowski's "taxicab geometry."