Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2000-06-14
Phys. Rev. E, 62 (2000) 7735
Physics
Condensed Matter
Statistical Mechanics
7 pages Revtex, 3 eps figures
Scientific paper
10.1103/PhysRevE.62.7735
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
Krapivsky Paul. L.
Majumdar Satya N.
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