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Random currents for signed interactions in the one-dimensional heterogeneous Ising model

Edward Athaide, Maciej Głuchowski, Jonas Köppl, Georg Menz

Abstract

The random current representation has been very successful in analyzing the ferromagnetic Ising model. In this article, we extend the method to the non-ferromagnetic case. To show its usefulness, we derive a covariance identity for the heterogeneous one-dimensional Ising model, which recovers the correct rate of exponential decay of covariances in the case of random i.i.d. interaction and external field.

Random currents for signed interactions in the one-dimensional heterogeneous Ising model

Abstract

The random current representation has been very successful in analyzing the ferromagnetic Ising model. In this article, we extend the method to the non-ferromagnetic case. To show its usefulness, we derive a covariance identity for the heterogeneous one-dimensional Ising model, which recovers the correct rate of exponential decay of covariances in the case of random i.i.d. interaction and external field.
Paper Structure (12 sections, 9 theorems, 95 equations)

This paper contains 12 sections, 9 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. The random current representation
  3. The model
  4. The transfer matrix method
  5. Main results
  6. The truncated chain
  7. Statement of main results
  8. Random current expansion for signed interactions
  9. Application to the truncated chain
  10. Corollaries of the matrix identity
  11. Calculations for the truncated chain
  12. Proof of Theorem \ref{['thm:matrix-identity']} using standard techniques of statistical mechanics

Key Result

Theorem 2.1

For the one-dimensional truncated chain described above, where $e_1$ is the first standard basis vector in $\mathbb{R}^2$, and

Theorems & Definitions (25)

  • Theorem 2.1
  • Remark 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Remark 2.6
  • Definition 3.1
  • Remark 3.2
  • Definition 3.4
  • Lemma 3.5: Switching lemma GHS70Aiz82
  • ...and 15 more