Height-offset variables and pinning at infinity for gradient Gibbs measures on trees

TL;DR

The paper develops a rigorous framework for height-offset variables (HOVs) to lift gradient Gibbs measures on Cayley trees to full Gibbs measures, focusing on free and height-period-2 GGMs under a finite-second-moment condition on the transfer operator. It proves existence of HOVs as $L^2$-limits of spherical averages, establishes that the HOVs have smooth Lebesgue densities and yield pinned measures with exponential localization, and shows that pinning at infinity disrupts translation symmetry, tree-indexed Markov properties, and extremality. The results are supported by martingale techniques, tail decompositions, and infinite product representations, and are illustrated with SOS and discrete Gaussian models. Overall, the work clarifies the localization-delocalization interplay on trees and highlights the nuanced behavior of liftable GGMs under pinning, with implications for constructing Gibbs measures from gradient data on non-Euclidean graphs.

Abstract

Height-offset variables (HOVs) provide a mechanism, known as "pinning at infinity", to lift gradient Gibbs measures (GGMs) - describing interface increments - to proper Gibbs measures that describe absolute heights. Starting from Sheffield's seminal framework, we study HOVs for nearest-neighbor integer-valued gradient models on regular trees, under broad classes of transfer operators requiring only finite second moments and without assuming convexity. We first establish the existence of HOVs as martingale limits, prove the infinite differentiability of their Lebesgue densities, and demonstrate exponential concentration for the associated pinned Gibbs measures. Next we uncover a fundamental trade-off, as the Gibbs measures arising by "pinning at infinity" paradoxically lose several desirable structural properties. We rigorously show that they lose tree-automorphism invariance, the tree-indexed Markov chain property, and extremality within the class of Gibbs measures. Our analysis relies on martingale theory, novel past- and future-tail decompositions, and infinite product representations for moment generating functions, and it applies to free GGMs, as well as to GGMs of height-period two.

Figures (2)

• Figure 1: An illustration for the recursion leading to Eq. \ref{['eq: SplitPoint']}. The graphics shows the ball $B_r$ of radius $r=3$ on the Cayley tree of order $d=2$ with some distinguished vertex $v_1 \in W_r$. The vertex $v_1$ is the unique vertex for which the function $\mathbbm{u}$ takes its maximal value $r$. The $d-1=1$ vertices filled with vertical lines are the vertices for which $\mathbbm{u}$ takes the value $r-1$. The $d(d-1)$ vertices filled with horizontal lines are the vertices for which $\mathbbm{u}$ takes the value $r-2$. The recursion stops here, as $r-3=0$.
• Figure 2: The situation on the binary tree $\mathbb{T}^2$. The choice of $v$ induces a splitting of $\mathbb{T}^d$ into the blue regular $d$-ary subtree rooted at $\rho$ and the red regular $d$-ary subtree of rooted at $v$. The tail-measurable random variables $\overleftarrow{H}^{\rho,0}$ and $\overrightarrow{H}^{v,0}$ are the limits of spherical averages in the past (the future, resp.). Finally, the map $l_v$ interchanges the roles of the past and the future.

Theorems & Definitions (32)

• Definition 1
• Theorem 1: Gibbs property of measures pinned at infinity
• Remark 1
• Definition 2: Spherical averages of height- and pinned gradient configurations
• Remark 2
• Definition 3: Free state of i.i.d-increments
• Theorem 2: Pinning at infinity of the free gradient state: existence, regularity of density, loss of translation invariance and of Markov property
• Definition 4: 2-height periodic GGM
• Theorem 3: Pinning at infinity for gradient Gibbs measures of height period $2$: existence, regularity of density, loss of translation invariance and loss of extremality
• Proposition 1: Martingale construction for the free state