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The Supercritical Loop O(1) and Random Current models: Uniqueness and Mixing

Ulrik Thinggaard Hansen, Frederik Ravn Klausen

Abstract

Much recent rigorous study of the classical ferromagnetic Ising model has been powered by its graphical representations, such as the random current and loop O(1) model (high temperature expansion). In this paper, we prove uniqueness of Gibbs measures and exponential ratio weak mixing for the loop O(1) and random current models corresponding to the supercritical Ising model on the hypercubic lattice $\Z^d$ in any dimension $d \geq 2$. The main technical innovation is to establish unique crossing events for conditional random-cluster measures by a delicate exploration coupling of Pisztora's coarse-graining method across scales. The results generalise to $q$-flow models and have natural applications for gradient measures of $\Z/q\mathbb{Z}$-gauge theories.

The Supercritical Loop O(1) and Random Current models: Uniqueness and Mixing

Abstract

Much recent rigorous study of the classical ferromagnetic Ising model has been powered by its graphical representations, such as the random current and loop O(1) model (high temperature expansion). In this paper, we prove uniqueness of Gibbs measures and exponential ratio weak mixing for the loop O(1) and random current models corresponding to the supercritical Ising model on the hypercubic lattice in any dimension . The main technical innovation is to establish unique crossing events for conditional random-cluster measures by a delicate exploration coupling of Pisztora's coarse-graining method across scales. The results generalise to -flow models and have natural applications for gradient measures of -gauge theories.

Paper Structure

This paper contains 20 sections, 32 theorems, 118 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Organisation of Paper and Proof Sketch for \ref{['proposition:unique_crossing_with_fa']}
  3. Open Problems.
  4. Setup, notation, and necessary basic properties
  5. On the use of constants
  6. Proofs of Main Theorems
  7. From unique crossings to unique measures: Proof of Theorem \ref{['thm:Unique_Loop']}
  8. Consequences for random currents: Proof of Theorem \ref{['thm:Free-mix']}
  9. Mixing, uniqueness of Gibbs measures and uniqueness of weak limits
  10. Pisztora's giant meets the boundary
  11. Exploration coupling
  12. Pisztora's giants touch a density of points on the boundary
  13. Unique FK-crossings conditioned on F_A: Proof of Proposition \ref{['proposition:unique_crossing_with_fa']}
  14. Catching via exploration
  15. Unique crossings conditioned on F_A
  16. ...and 5 more sections

Key Result

Theorem 1.1

For any $d\geq 2$ and $x>x_c,$ there exists a measure $\ell_{\mathbb{Z}^d,x}$ such that for any exhaustion $G_N\nearrow \mathbb{Z}^d$ and any $A_N\subseteq \partial_v G_N$ with $|A_N|$ even, Furthermore, $\ell_{\mathbb{Z}^d,x}$ is exponentially ratio weak mixing. In particular, $\ell_{\mathbb{Z}^d,x}$ is the unique Gibbs measure for the loop O($1$) model.

Figures (3)

  • Figure 1: Two root-to-leaf paths $\gamma,\gamma'\in \Gamma_n^A$ sharing a common prefix down to $v_k$ at generation $k$. By condition $(ii)$, the non-open events on $\gamma \setminus \gamma'$ and $\gamma'\setminus \gamma$ are independent, so $\operatorname{Cov}[\mathbf{1}_{\gamma\in\mathfrak{C}},\mathbf{1}_{\gamma'\in\mathfrak{C}}]\leq \nu[\gamma\cap\gamma'\in\mathfrak{C}]$.
  • Figure 2: One step of the dyadic subdivision process. On the left: Giant clusters existing under free boundary conditions on each smaller box in various shades of blue. In the middle: The giant component under free boundary conditions on the larger box in red and orange. On the right: The blue giants are so large that, under the increasing coupling, they must be connected to the red giant via red edges. Due to planar limitations of the graphical presentation, we have elected not to indicate that all giants are connected to the boundary.
  • Figure 3: Schematic of the strategy in the proof of \ref{['The_Catching_Prop']}. In every annulus of size $\log(N)$, there is a giant cluster in the annulus and with probability tending to $1,$ an $\varepsilon$-fraction of the points in $A^k$ connect to this giant cluster.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • proof : Proof of \ref{['thm:Free-mix']}.
  • Lemma 4.1
  • proof
  • ...and 47 more