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On the Effect of Bottlenecks in Block Spin Models

Isabel Lammers, Matthias Löwe

Abstract

We study a bottleneck spin model with $N$ spins, split into two Curie-Weiss models at low temperature with a bottleneck between them. We propose multiple ways of how to realize such a bottleneck and study its influence on the phase transition in the thermodynamic limit $N \to \infty$. In all versions of this model we prove the existence of a threshold that determines whether or not the presence of the bottleneck is felt in the phase transition. This threshold depends on the size of the bottleneck and the interaction strength through it.

On the Effect of Bottlenecks in Block Spin Models

Abstract

We study a bottleneck spin model with spins, split into two Curie-Weiss models at low temperature with a bottleneck between them. We propose multiple ways of how to realize such a bottleneck and study its influence on the phase transition in the thermodynamic limit . In all versions of this model we prove the existence of a threshold that determines whether or not the presence of the bottleneck is felt in the phase transition. This threshold depends on the size of the bottleneck and the interaction strength through it.
Paper Structure (8 sections, 16 theorems, 192 equations)

This paper contains 8 sections, 16 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Models and main results
  3. A perfect matching
  4. A diluted matching
  5. A 3-block model
  6. Proof of Theorem \ref{['theo:theo1']}
  7. Proof of Theorem \ref{['theo:theo2']}
  8. Proof of Theorems \ref{['thm:Blockspin']} and \ref{['thm:PhaseTransition']}

Key Result

Theorem 2.1

Consider the two-block spin model with $\alpha_N>0$ such that $\alpha_N\downarrow 0$.

Theorems & Definitions (33)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • proof
  • ...and 23 more