Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of gradient Gibbs measures on regular trees which are not translation invariant

Florian Henning, Christof Kuelske

Abstract

We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.

Existence of gradient Gibbs measures on regular trees which are not translation invariant

Abstract

We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.

Paper Structure

This paper contains 25 sections, 20 theorems, 85 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main statement
  3. Ideas of the proof
  4. Definitions
  5. Height configurations, Markov chains and translations on the Cayley tree
  6. Gibbs measures and gradient Gibbs measures
  7. Two-layer hidden Markov model construction in automorphism non invariant setup
  8. Background on relation between boundary laws and Gibbs measures
  9. Height-periodic boundary laws
  10. A two-step construction of gradient Gibbs measures
  11. The gradient Gibbs property
  12. Delocalization of possibly nonhomogeneous gradient Gibbs measures for homogeneous specifications
  13. Existence theory for non-invariant GGMs of arbitrarily large periods $q$ via pseudo-unstable manifold theorem
  14. Main results
  15. Boundary laws in forward and backwards direction
  16. ...and 10 more sections

Key Result

Theorem 1

Let $(Q_b)_{b \in L}$ be any family of transfer operators such that there is some $\omega \in \Omega$ with Then for the Markov specification $\gamma$ associated to $(Q_b)_{b \in L}$ we have:

Figures (3)

  • Figure 1: On the Cayley tree of order $d=5$, period $q=16$ and the inverse square model (see Section \ref{['sec: Applications']} below) the non-hyperbolic case occurs at an exceptional value of $a=\frac{1536}{73 \pi ^2} \approx 2.132$. The bars mark the values of $\hat{Q}(k)/\hat{Q}(0)$ at $k=j \frac{\pi}{8}$, where $j=1, \ldots,8$. The third bar from the left at $k=\frac{3\pi}{8}$ hits the upper dashed horizontal line marking the threshold $\frac{1}{d}=\frac{1}{5}$ and hence represents a neutral eigenvalue of $\text{D}S_q[eq]$. To satisfy the hypothesis of Theorem \ref{['thm: existence']}, we may put $\tau=\frac{5}{4}$ and slightly shift the dashed lines away from the horizontal axis to the solid horizontal lines at $\pm \frac{1}{4}$.
  • Figure 2: In the case $q=2$ the simplex is a unit interval. The picture shows the graph of the map $S_2$ where $d=3$ and $Q$ is such that $Q^2(\bar{0})/Q^2(\bar{1})=4>\frac{d+1}{d-1}$. The global unstable manifold around the equidistribution (center red point) is the cyan open interval between the two nontrivial fixed points of $S_2$. The solid line between the two graphs represents three backwards-iterations of the map $S_2$, starting at about $0.72$.
  • Figure 3: The graphs of the function $\hat{Q}(\cdot)/\hat{Q}(0)$ (see Theorem \ref{['thm: existence']}) for the two models at different parameter values.

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1: Theorem 3.2 in Z83
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • ...and 33 more