ISSN:
1572-9613
Keywords:
Percolation
;
minimal spanning tree
;
free energy
;
critical value
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1023092317419
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