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.
Categories: cond-mat.stat-mech