ISSN:
1572-9613
Schlagwort(e):
Percolation
;
minimal spanning tree
;
free energy
;
critical value
Quelle:
Springer Online Journal Archives 1860-2000
Thema:
Physik
Notizen:
Abstract Consider a random set $$V_n $$ of points in the box [n, −n) d , generated either by a Poisson process with density p or by a site percolation process with parameter p. We analyze the empirical distribution function F n of the lengths of edges in a minimal (Euclidean) spanning tree $$T_n $$ on $$V_n$$ . We express the limit of F n, as n → ∞, in terms of the free energies of a family of percolation processes derived from $$V_n$$ by declaring two points to be adjacent whenever they are closer than a prescribed distance. By exploring the singularities of such free energies, we show that the large-n limits of the moments of F n are infinitely differentiable functions of p except possibly at values belonging to a certain infinite sequence (p c(k): k ≥ 1) of critical percolation probabilities. It is believed that, in two dimensions, these limiting moments are twice differentiable at these singular values, but not thrice differentiable. This analysis provides a rigorous framework for the numerical experimentation of Dussert, Rasigni, Rasigni, Palmari, and Llebaria, who have proposed novel Monte Carlo methods for estimating the numerical values of critical percolation probabilities.
Materialart:
Digitale Medien
URL:
http://dx.doi.org/10.1023/A:1023092317419
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