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Macroscopic behavior of Lipschitz random surfaces

Piet Lammers, Martin Tassy

Abstract

The motivation for this article is to derive strict convexity of the surface tension for Lipschitz random surfaces, that is, for models of random Lipschitz functions from $\mathbb Z^d$ to $\mathbb Z$ or $\mathbb R$. An essential innovation is that random surface models with long- and infinite-range interactions are included in the analysis. More specifically, we cover at least: uniformly random graph homomorphisms from $\mathbb Z^d$ to a $k$-regular tree for any $k\geq 2$ and Lipschitz potentials which satisfy the FKG lattice condition. The latter includes perturbations of dimer- and six-vertex models and of Lipschitz simply attractive potentials introduced by Sheffield. The main result is that we prove strict convexity of the surface tension -- which implies uniqueness for the limiting macroscopic profile -- if the model of interest is monotone in the boundary conditions. This solves a conjecture of Menz and Tassy, and answers a question posed by Sheffield. Auxiliary to this, we prove several results which may be of independent interest, and which do not rely on the model being monotone. This includes existence and topological properties of the specific free energy, as well as a characterization of its minimizers. We also prove a general large deviations principle which describes both the macroscopic profile and the local statistics of the height functions. This work is inspired by, but independent of, Random Surfaces by Sheffield.

Macroscopic behavior of Lipschitz random surfaces

Abstract

The motivation for this article is to derive strict convexity of the surface tension for Lipschitz random surfaces, that is, for models of random Lipschitz functions from to or . An essential innovation is that random surface models with long- and infinite-range interactions are included in the analysis. More specifically, we cover at least: uniformly random graph homomorphisms from to a -regular tree for any and Lipschitz potentials which satisfy the FKG lattice condition. The latter includes perturbations of dimer- and six-vertex models and of Lipschitz simply attractive potentials introduced by Sheffield. The main result is that we prove strict convexity of the surface tension -- which implies uniqueness for the limiting macroscopic profile -- if the model of interest is monotone in the boundary conditions. This solves a conjecture of Menz and Tassy, and answers a question posed by Sheffield. Auxiliary to this, we prove several results which may be of independent interest, and which do not rely on the model being monotone. This includes existence and topological properties of the specific free energy, as well as a characterization of its minimizers. We also prove a general large deviations principle which describes both the macroscopic profile and the local statistics of the height functions. This work is inspired by, but independent of, Random Surfaces by Sheffield.

Paper Structure

This paper contains 77 sections, 75 theorems, 274 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preface
  3. Description of the main results
  4. Ideas and strategy of the proof
  5. Open questions
  6. The thermodynamical formalism
  7. The gradient formalism
  8. Height functions
  9. Subsets of $\mathbb Z^d$
  10. $\sigma$-Algebras and random fields
  11. The topology of (weak) local convergence
  12. Shift-invariance and ergodicity
  13. Specifications
  14. Monotonicity
  15. The Lipschitz property
  16. ...and 62 more sections

Key Result

Theorem 1

Let $\Phi$ denote a potential which decomposes as described above, and such that the induced specification $\gamma^\Phi$ is monotone.

Figures (8)

  • Figure 1: Limiting behavior of the $5$-vertex model for different parameters
  • Figure 2: This figure shows the boundaries of the upper level sets of the horocylic height function (presented in Subsection \ref{['subsection:applications_tree']}) of a random $\mathcal{T}_3$-valued graph homomorphism. The boundary conditions resemble the Aztec diamond for domino tilings. The simulation hints at the presence of an arctic circle, alongside the limit shape which we prove appears inside.
  • Figure 3: The random truncation for $E=\mathbb R$. The randomly truncated sample $\psi$ remains between $\phi^-_n-2\varepsilon$ and $\phi^+_n+2\varepsilon$.
  • Figure 4: The washboard and the function $f:\partial'D\cup D'\to\mathbb R$
  • Figure 5: The sets $D"\subset D'\subset D\subset\mathbb R^d$, and the sets $\Sigma^*\subset\Sigma$ of simplices of scale $\varepsilon_2$
  • ...and 3 more figures

Theorems & Definitions (168)

  • Theorem : strict convexity of the surface tension
  • Theorem : minimizers of the specific free energy
  • Theorem : variational principle
  • Definition 3.1: local Lipschitz constraint
  • Definition 3.2: $q$-Lipschitz
  • Definition 3.3: $U_q$, $\|\cdot\|_q$
  • Definition 3.4: $\|\cdot\|$-Lipschitz
  • Definition 3.5: strong interaction, $\mathcal{S}_\mathcal{L}$
  • Definition 3.6: summability
  • Definition 3.7: amenability
  • ...and 158 more