1207.7108 (Yevgeniy Kovchegov et al.)

Horton self-similarity of Kingman's coalescent process    [PDF]

Yevgeniy Kovchegov, Ilia Zaliapin

The paper establishes Horton self-similarity for Kingman's coalescent process and level-set tree of a white noise. The proof is based on a Smoluchowski-type system of ordinary differential equations for the number of Horton-Strahler branches in a tree that represents Kingman's coalescent. This approach allows to establish a weak form of the Horton law, namely the existence of a limit of (N_k/N)^{1/k} for the number N_k of branches of Horton-Strahler order k in a system of size N, as we let N \rightarrow \infty and then k \rightarrow \infty. We conjecture, based on numerical observations, that the Kingman's coalescent is also Horton self-similar in regular strong sense with Horton exponent R = lim_{k \rightarrow\infty} lim_{N \rightarrow\infty} (N_{k+1}/N_k)=0.328533... and asymptotically Tokunaga self-similar. Finally, we establish equivalence between the trees of a finite Kingman's coalescent and level-set trees of a discrete white noise.

View original: http://arxiv.org/abs/1207.7108