Macroscopic loops in the loop O(n) model via the XOR trick
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2020
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| author | Crawford, Nicholas Glazman, Alexander Harel, Matan Peled, Ron |
| author_facet | Crawford, Nicholas Glazman, Alexander Harel, Matan Peled, Ron |
| contents | The loop $O(n)$ model is a family of probability measures on collections of non-intersecting loops on the hexagonal lattice, parameterized by a loop-weight $n$ and an edge-weight $x$. Nienhuis predicts that, for $0 \leq n \leq 2$, the model exhibits two regimes separated by $x_c(n) = 1/\sqrt{2 + \sqrt{2-n}}$: when $x < x_c(n)$, the loop lengths have exponential tails, while, when $x \geq x_c(n)$, the loops are macroscopic.
In this paper, we prove three results regarding the existence of long loops in the loop $O(n)$ model:
- In the regime $(n,x) \in [1,1+δ) \times (1- δ, 1]$ with $δ>0$ small, a configuration sampled from a translation-invariant Gibbs measure will either contain an infinite path or have infinitely many loops surrounding every face. In the subregime $n \in [1,1+δ)$ and $x \in (1-δ,1/\sqrt{n}]$ our results further imply Russo--Seymour--Welsh theory. This is the first proof of the existence of macroscopic loops in a positive area subset of the phase diagram.
- Existence of loops whose diameter is comparable to that of a finite domain whenever $n=1, x \in (1,\sqrt{3}]$; this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice.
- Existence of non-contractible loops on a torus when $n \in [1,2], x=1$.
The main ingredients of the proof are: (i) the `XOR trick': if $ω$ is a collection of short loops and $Γ$ is a long loop, then the symmetric difference of $ω$ and $Γ$ necessarily includes a long loop as well; (ii) a reduction of the problem of finding long loops to proving that a percolation process on an auxiliary planar graph, built using the Chayes--Machta and Edwards--Sokal geometric expansions, has no infinite connected components; and (iii) a recent result on the percolation threshold of Benjamini--Schramm limits of planar graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_11977 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Macroscopic loops in the loop O(n) model via the XOR trick Crawford, Nicholas Glazman, Alexander Harel, Matan Peled, Ron Probability Mathematical Physics 60K35, 82B05 82B20, 82B26, 82B27, 60C05 The loop $O(n)$ model is a family of probability measures on collections of non-intersecting loops on the hexagonal lattice, parameterized by a loop-weight $n$ and an edge-weight $x$. Nienhuis predicts that, for $0 \leq n \leq 2$, the model exhibits two regimes separated by $x_c(n) = 1/\sqrt{2 + \sqrt{2-n}}$: when $x < x_c(n)$, the loop lengths have exponential tails, while, when $x \geq x_c(n)$, the loops are macroscopic. In this paper, we prove three results regarding the existence of long loops in the loop $O(n)$ model: - In the regime $(n,x) \in [1,1+δ) \times (1- δ, 1]$ with $δ>0$ small, a configuration sampled from a translation-invariant Gibbs measure will either contain an infinite path or have infinitely many loops surrounding every face. In the subregime $n \in [1,1+δ)$ and $x \in (1-δ,1/\sqrt{n}]$ our results further imply Russo--Seymour--Welsh theory. This is the first proof of the existence of macroscopic loops in a positive area subset of the phase diagram. - Existence of loops whose diameter is comparable to that of a finite domain whenever $n=1, x \in (1,\sqrt{3}]$; this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. - Existence of non-contractible loops on a torus when $n \in [1,2], x=1$. The main ingredients of the proof are: (i) the `XOR trick': if $ω$ is a collection of short loops and $Γ$ is a long loop, then the symmetric difference of $ω$ and $Γ$ necessarily includes a long loop as well; (ii) a reduction of the problem of finding long loops to proving that a percolation process on an auxiliary planar graph, built using the Chayes--Machta and Edwards--Sokal geometric expansions, has no infinite connected components; and (iii) a recent result on the percolation threshold of Benjamini--Schramm limits of planar graphs. |
| title | Macroscopic loops in the loop O(n) model via the XOR trick |
| topic | Probability Mathematical Physics 60K35, 82B05 82B20, 82B26, 82B27, 60C05 |
| url | https://arxiv.org/abs/2001.11977 |