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Theory of minimum spanning trees. I. Mean-field theory and strongly disordered<?xpp qa?> spin-glass model

Theory of minimum spanning trees. I. Mean-field theory and strongly disordered... The minimum spanning tree (MST) is a combinatorial optimization problem: given a connected graph with a real weight (“cost”) on each edge, find the spanning tree that minimizes the sum of the total cost of the occupied edges. We consider the random MST, in which the edge costs are (quenched) independent random variables. There is a strongly disordered spin-glass model due to Newman and Stein Phys. Rev. Lett. 72 , 2286 ( 1994 ) , which maps precisely onto the random MST. We study scaling properties of random MSTs using a relation between Kruskal’s greedy algorithm for finding the MST, and bond percolation. We solve the random MST problem on the Bethe lattice (BL) with appropriate wired boundary conditions and calculate the fractal dimension D = 6 of the connected components. Viewed as a mean-field theory, the result implies that on a lattice in Euclidean space of dimension d , there are of order W d − D large connected components of the random MST inside a window of size W , and that d = d c = D = 6 is a critical dimension. This differs from the value 8 suggested by Newman and Stein. We also critique the original argument for 8, and provide an improved scaling argument that again yields d c = 6 . The result implies that the strongly disordered spin-glass model has many ground states for d > 6 , and only of order one below six. The results for MSTs also apply on the Poisson-weighted infinite tree, which is a mean-field approach to the continuum model of MSTs in Euclidean space, and is a limit of the BL. In a companion paper we develop an ε = 6 − d expansion for the random MST on critical percolation clusters. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review E American Physical Society (APS)

Theory of minimum spanning trees. I. Mean-field theory and strongly disordered<?xpp qa?> spin-glass model

Physical Review E , Volume 81 (2) – Feb 1, 2010
16 pages

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Publisher
American Physical Society (APS)
Copyright
Copyright © 2010 The American Physical Society
ISSN
1550-2376
DOI
10.1103/PhysRevE.81.021130
pmid
20365553
Publisher site
See Article on Publisher Site

Abstract

The minimum spanning tree (MST) is a combinatorial optimization problem: given a connected graph with a real weight (“cost”) on each edge, find the spanning tree that minimizes the sum of the total cost of the occupied edges. We consider the random MST, in which the edge costs are (quenched) independent random variables. There is a strongly disordered spin-glass model due to Newman and Stein Phys. Rev. Lett. 72 , 2286 ( 1994 ) , which maps precisely onto the random MST. We study scaling properties of random MSTs using a relation between Kruskal’s greedy algorithm for finding the MST, and bond percolation. We solve the random MST problem on the Bethe lattice (BL) with appropriate wired boundary conditions and calculate the fractal dimension D = 6 of the connected components. Viewed as a mean-field theory, the result implies that on a lattice in Euclidean space of dimension d , there are of order W d − D large connected components of the random MST inside a window of size W , and that d = d c = D = 6 is a critical dimension. This differs from the value 8 suggested by Newman and Stein. We also critique the original argument for 8, and provide an improved scaling argument that again yields d c = 6 . The result implies that the strongly disordered spin-glass model has many ground states for d > 6 , and only of order one below six. The results for MSTs also apply on the Poisson-weighted infinite tree, which is a mean-field approach to the continuum model of MSTs in Euclidean space, and is a limit of the BL. In a companion paper we develop an ε = 6 − d expansion for the random MST on critical percolation clusters.

Journal

Physical Review EAmerican Physical Society (APS)

Published: Feb 1, 2010

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