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UBC Theses and Dissertations

Uniform spanning forest on the integer lattice with drift in one coordinate. Martínez Dibene, Guillermo Elías

Abstract

The purpose of this thesis is to investigate the Uniform Spanning Forest (USF) in the nearestneighbour integer lattice Z^{d+1} = ZxZ^d with the family of conductances c((n, x), (n', x')) = e^{λ max(n,n')}, for all (n, x) ~ (n', x'), where ~ stands here for adjacency of vertices and λ > 0 is a fixed parameter. The random walk corresponding to this assignment of conductances resembles a discrete version of Brownian motion with drift. These conductances are not bounded, neither from below nor from above. This entails that many results known in the literature are not applicable. Our results include: 1. Estimate, up to multiplicative constants, the Green's function of this network. 2. Both the Wired and Free uniform spanning forest measures coincide for these assignment of conductances via a coupling argument. Call USF the resulting measure. 3. For d = 1; 2 we will show that USF consists of a tree; for d ≥ 3; it consists of infinitely many infinite trees. 4. For every d; every component of USF is one-ended. 5. For d ≥ 3; there exists a metric η on Z^{d+1} such that the probability that z, z' belong to the same USF-component is bounded above and below by multiplicative constants of η(z - z')^{-(d-2)}: The upper bound is then generalised to the probability that all the vertices in a finite set should belong to the same component. 6. Collapse every tree in USF to a point and denote by D(z, z') the number of edges that separate the USF-components at z and z': Almost surely, D(z, z') = U+2308 (d-2)/4 U+2309

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