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Updated: Feb 26, 2026

Setting Limits on Supersymmetry Using Simplified Models
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Correspondence between spanning trees and the Ising model on a square lattice.

G M Viswanathan1

  • 1Department of Physics and National Institute of Science and Technology of Complex Systems, Universidade Federal do Rio Grande do Norte, 59078-970 Natal-RN, Brazil.

Physical Review. E
|July 16, 2017
PubMed
Summary

Statistical physics reveals a connection between Ising model partition functions and spanning tree counts. This relationship, generalized to all temperatures, links these quantities via the Mahler measure and random walk functions.

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Area of Science:

  • Statistical Physics
  • Graph Theory
  • Mathematical Physics

Background:

  • Investigates the link between partition functions in statistical mechanics and graph enumeration problems.
  • Focuses on the relationship between spanning trees and the Ising model partition function on a square lattice.

Purpose of the Study:

  • To generalize the known connection between the Ising model partition function at critical temperature and spanning tree counts to all real temperatures.
  • To establish a precise mathematical relationship between the Ising model partition function and the spanning tree generating function.

Main Methods:

  • Utilizes the spanning tree generating function T(z) and the Ising model partition function Z(K).
  • Proves the identity [Z(K)sech2K]^2 = k exp[T(k)], where k = 2tanh(2K)sech(2K).
  • Connects these quantities through the Mahler measure and the random walk structure function.

Main Results:

  • Demonstrates that the relationship between the Ising model partition function and spanning tree counts holds for all real temperatures, not just the critical temperature.
  • Identifies the Mahler measure as a unifying concept linking the partition function and spanning tree generating function.
  • Shows that this correspondence does not directly extend to nonplanar lattices.

Conclusions:

  • Establishes a novel, temperature-dependent connection between equilibrium statistical mechanics and graph theory on the square lattice.
  • Highlights the role of the Mahler measure and random walk structure function in bridging these fields.
  • Suggests limitations of this specific correspondence for more complex lattice structures.