Extremal Paths on a Random Cayley Tree

We investigate the statistics of extremal path(s) (both the shortest and the longest) from the root to the bottom of a Cayley tree. The lengths of the edges are assumed to be independent identically distributed random variables drawn from a distribution \rho(l). Besides, the number of branches from any node is also random. Exact results are derived for arbitrary distribution \rho(l). In particular, for the binary {0,1} distribution \rho(l)=p\delta_{l,1}+(1-p)\delta_{l,0}, we show that as p increases, the minimal length undergoes an unbinding transition from a `localized' phase to a `moving' phase at the critical value, p=p_c=1-b^{-1}, where b is the average branch number of the tree. As the height n of the tree increases, the minimal length saturates to a finite constant in the localized phase (p<p_c), but increases linearly as v_{min}(p)n in the moving phase (p>p_c) where the velocity v_{min}(p) is determined via a front selection mechanism. At p=p_c, the minimal length grows with n in an extremely slow double logarithmic fashion. The length of the maximal path, on the other hand, increases linearly as v_{max}(p)n for all p. The maximal and minimal velocities satisfy a general duality relation, v_{min}(p)+v_{max}(1-p)=1, which is also valid for directed paths on finite-dimensional lattices.