Random-cluster measures and uniform spanning trees

DOI10.1016/0304-4149(95)00042-6zbMath0840.60089OpenAlexW2065648252MaRDI QIDQ1904547

Olle Häggström

Publication date: 18 June 1996

Published in: Stochastic Processes and their Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0304-4149(95)00042-6

zbMATH Keywords

bond percolationPotts modelsrandom cluster measure

Mathematics Subject Classification ID

Interacting random processes; statistical mechanics type models; percolation theory (60K35) Combinatorial probability (60C05) Convergence of probability measures (60B10)

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Indistinguishability of trees in uniform spanning forests ⋮ Conformal invariance of planar loop-erased random walks and uniform spanning trees. ⋮ Determinantal probability measures ⋮ Minimal spanning forests ⋮ The random-cluster model on a homogeneous tree ⋮ Spectral dimension and random walks on the two dimensional uniform spanning tree ⋮ Percolation beyond \(\mathbb Z^{d}\): the contributions of Oded Schramm ⋮ Hölder regularity and dimension bounds for random curves ⋮ Towards a theory of negative dependence ⋮ Coloring percolation clusters at random. ⋮ Decomposition and convergence for tree martingales ⋮ The component graph of the uniform spanning forest: transitions in dimensions \(9,10,11,\ldots\) ⋮ UNIFORM SPANNING FORESTS OF PLANAR GRAPHS ⋮ Sandpile models ⋮ Universality of high-dimensional spanning forests and sandpiles ⋮ Invariant coupling of determinantal measures on sofic groups ⋮ Negative dependence and the geometry of polynomials ⋮ Uniform spanning forests on biased Euclidean lattices ⋮ Scaling limits of loop-erased random walks and uniform spanning trees ⋮ Spanning forests and the \(q\)-state Potts model in the limit \(q \to 0\) ⋮ Entropy for translation-invariant random-cluster measures

Cites Work

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• Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances
• Density and uniqueness in percolation
• Choosing a spanning tree for the integer lattice uniformly
• Gibbs measures and phase transitions
• Comparison and disjoint-occurrence inequalities for random-cluster models.
• Discontinuity of the magnetization in one-dimensional \(1/| x-y| ^ 2\) Ising and Potts models.
• The Random Walk Construction of Uniform Spanning Trees and Uniform Labelled Trees