Minimum spanning trees and random resistor networks in d dimensions

TL;DR

This paper investigates the universal properties and scaling behaviors of minimum spanning trees in d-dimensional lattices and Euclidean spaces, revealing connections to percolation theory and resistor networks.

Contribution

It establishes a universal correction exponent for minimum spanning trees in various dimensions and links it to percolation correlation length exponents, providing new theoretical insights.

Findings

Universal correction exponent theta < 0 depending on dimension

Scaling relation theta = -1/nu linking to percolation

Universality class includes Steiner and traveling salesman problems in 2D

Abstract

We consider minimum-cost spanning trees, both in lattice and Euclidean models, in d dimensions. For the cost of the optimum tree in a box of size L, we show that there is a correction of order L^theta, where theta < 0 is a universal d-dependent exponent. There is a similar form for the change in optimum cost under a change in boundary condition. At non-zero temperature T, there is a crossover length xi approx equal to T^{-nu}, such that on length scales larger than xi, the behavior becomes that of uniform spanning trees. There is a scaling relation theta=-1/nu, and we provide several arguments that show that nu and -1/theta both equal nu_perc, the correlation length exponent for ordinary percolation in the same dimension d, in all dimensions d > 1. The arguments all rely on the close relation of Kruskal's greedy algorithm for the minimum spanning tree, percolation, and (for some arguments) random resistor networks. The scaling of the entropy and free energy at small non-zero T, and hence of the number of near-optimal solutions, is also discussed. We suggest that the Steiner tree problem is in the same universality class as the minimum spanning tree in all dimensions, as is the traveling salesman problem in two dimensions. Hence all will have the same value of theta=-3/4 in two dimensions.