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Exponents and bounds for uniform spanning trees in d dimensions.

N Read1

  • 1Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 28, 2004
PubMed
Summary

Uniform spanning trees, a statistical model, are analyzed using graph theory and Grassmann integrals. This study provides exact exponents bounding the probability of branch proximity in large lattices.

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Area of Science:

  • Statistical Physics
  • Graph Theory
  • Mathematical Physics

Background:

  • Uniform spanning trees (USTs) are fundamental objects in statistical mechanics, representing random subgraphs of a lattice.
  • Understanding the geometric properties of USTs, such as branch behavior, is crucial for characterizing their statistical behavior.

Purpose of the Study:

  • To investigate the power-law decay of the probability that k distinct branches of a UST pass close to two distinct points.
  • To derive exact exponents that characterize this probability as the lattice size approaches infinity.

Main Methods:

  • Utilizing the Laplacian matrix of the graph and employing Grassmann integrals.
  • Applying techniques from random matrix theory and statistical field theory.

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Main Results:

  • Obtained exact exponents that bound the power-law decay of the probability of branch proximity.
  • Established theoretical bounds for the behavior of USTs in large-scale lattices.

Conclusions:

  • The study provides precise mathematical insights into the spatial distribution of branches in uniform spanning trees.
  • The derived exponents offer valuable tools for analyzing the large-scale geometric properties of these random structures.