A Variational Approach to Branching Random Walk in Random Environment

This paper considers an infinite system of particles on the integers $\mathbb{Z}$ that: (1) step to the right with a random delay, and (2) split or die along the way according to a random law depending on their position. The exponential growth rate of the particle density is computed in the long time limit in the form of a variational formula that can be solved explicitly. The result reveals two phase transitions associated with localization vs. delocalization and survival vs. extinction. In addition, the system exhibits an intermittency effect. Greven and den Hollander considered the more difficult situation where the particles may step both to the left and right, but the analysis of the phase diagram was less complete