Branching annihilating random walk on random regular graphs

The branching annihilating random walk is studied on a random graph whose sites have a uniform number of neighbors (z). The Monte Carlo simulations in agreement with the generalized mean-field analysis indicate that the concentration decreases linearly with the branching rate for z greater than or equal to4, while the coefficient of the linear term becomes zero if z=3. These properties are described by a modified mean-field theory taking explicitly into consideration the probability of mutual annihilation of the parent and its offspring particles using the returning features of a single walker on the same graph.