Self-Avoiding Random Walks on Homogeneous Trees

#### E. Lebensztayn, F.P. Machado, M.Z. Martinez

2006, v.12, Issue 4, 735-745

ABSTRACT

We consider the following discrete-time particle system on a homogeneous tree of degree $(d + 1)$. At time zero, there is one particle at each vertex of the tree; only one of them is active, the others being inactive. Active particles perform independent discrete-time self-avoiding random walks with probability of disappearance $(1 - p)$ at each moment. An inactive particle becomes active once its vertex is hit by an active particle. We prove a phase transition result for this model, by stating explicit bounds for the critical probability. The criticality is with respect to the positivity of the probability of the event that there are active particles at all times. From the stated bounds, we conclude that the critical probability equals $1/2 + \theta(1/d)$.

Keywords: homogeneous tree,phase transition,self-avoiding walk