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Continuous eigenfunctions of the transfer operator for Dyson models

Anders Johansson, Anders Öberg, Mark Pollicott

TL;DR

This work proves the existence of a continuous eigenfunction for the transfer operator associated with long-range Dyson-type potentials under subcritical temperatures and square-summable variations of the one-point potential. The authors deploy a FK-Ising/random-cluster representation, casting the eigenfunction as a Radon-Nikodym derivative between two Gibbs measures and proving uniform convergence of finite-n likelihood ratios via percolation tail bounds. As a key corollary, they establish the existence of a continuous eigenfunction for Dyson potentials J(k)=β k^{-α} in the subcritical regime for α>3/2, thereby extending regularity results beyond summable variation. The results connect Gibbsian structure, g-measures/Doeblin theory, and percolation phenomena in long-range Ising models, with potential implications for statistical mechanics and ergodic theory of infinite-range systems.

Abstract

In this article we address a well known problem at the intersection of ergodic theory and statistical mechanics. We prove that there exists a continuous eigenfunction for the transfer operator corresponding to pair potentials that satisfy a square summability condition on the variations, when the inverse temperature is subcritical. As a corollary we obtain a continuous eigenfunction for the classical Dyson model, with interactions $\J(k)=β\, k^{-α}$, $k\ge1$, in the whole subcritical regime $β<β_c$ for which the parameter $α$ is greater than $3/2$.

Continuous eigenfunctions of the transfer operator for Dyson models

TL;DR

This work proves the existence of a continuous eigenfunction for the transfer operator associated with long-range Dyson-type potentials under subcritical temperatures and square-summable variations of the one-point potential. The authors deploy a FK-Ising/random-cluster representation, casting the eigenfunction as a Radon-Nikodym derivative between two Gibbs measures and proving uniform convergence of finite-n likelihood ratios via percolation tail bounds. As a key corollary, they establish the existence of a continuous eigenfunction for Dyson potentials J(k)=β k^{-α} in the subcritical regime for α>3/2, thereby extending regularity results beyond summable variation. The results connect Gibbsian structure, g-measures/Doeblin theory, and percolation phenomena in long-range Ising models, with potential implications for statistical mechanics and ergodic theory of infinite-range systems.

Abstract

In this article we address a well known problem at the intersection of ergodic theory and statistical mechanics. We prove that there exists a continuous eigenfunction for the transfer operator corresponding to pair potentials that satisfy a square summability condition on the variations, when the inverse temperature is subcritical. As a corollary we obtain a continuous eigenfunction for the classical Dyson model, with interactions , , in the whole subcritical regime for which the parameter is greater than .
Paper Structure (12 sections, 7 theorems, 65 equations)

This paper contains 12 sections, 7 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. The proofs of Theorem \ref{['thm:main']} and Theorem \ref{['main']}
  3. Preliminaries
  4. Potentials
  5. Graphs
  6. The eigenfunction as a Radon-Nikodym derivative
  7. The FK-Ising model
  8. Proof of Theorem \ref{['thm:main']}
  9. The cut
  10. The factorised representation
  11. The conclusion in the proof of Theorem \ref{['thm:main']}
  12. Proof that Theorem \ref{['thm:main']} implies Theorem \ref{['main']}

Key Result

Theorem 1

For $x\in X$, define $r(x) = \sum_{n=0}^\infty r_n x_n$. If $\nu\in\mathscr{M}_\phi$ and then there is a strictly positive continuous eigenfunction $h(x)\in C(X)$ of $\mathscr{L}=\mathscr{L}_\phi$ such that $\mathscr{L} h = \lambda h$.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Corollary 3
  • Remark 6
  • Remark 7
  • ...and 8 more