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2019 | OriginalPaper | Chapter

Lectures on the Spin and Loop O(n) Models

Authors : Ron Peled, Yinon Spinka

Published in: Sojourns in Probability Theory and Statistical Physics - I

Publisher: Springer Singapore

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Abstract

The classical spin O(n) model is a model on a d-dimensional lattice in which a vector on the \((n-1)\)-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact ferromagnetically via their inner product. Special cases include the Ising model (\(n=1\)), the XY model (\(n=2\)) and the Heisenberg model (\(n=3\)). We discuss questions of long-range order and decay of correlations in the spin O(n) model for different combinations of the lattice dimension d and the number of spin components n.
The loop O(n) model is a model for a random configuration of disjoint loops. We discuss its properties on the hexagonal lattice. The model is parameterized by a loop weight \(n\ge 0\) and an edge weight \(x\ge 0\). Special cases include self-avoiding walk (\(n=0\)), the Ising model (\(n=1\)), critical percolation (\(n=x=1\)), dimer model (\(n=1,x=\infty \)), proper 4-coloring (\(n=2, x=\infty )\), integer-valued (\(n=2\)) and tree-valued (integer \(n>=3\)) Lipschitz functions and the hard hexagon model (\(n=\infty \)). The object of study in the model is the typical structure of loops. We review the connection of the model with the spin O(n) model and discuss its conjectured phase diagram, emphasizing the many open problems remaining.

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Footnotes
1
Exponential decay is stated in these references in the infinite-volume limit, but is derived as a consequence of a finite-volume criterion and is thus implied, as the infinite-volume measure is unique, also in finite volume.
 
2
A related intuition was mentioned earlier by Herring and Kittel [68, Footnote 8a].
 
3
In fact, more is true, conditioned on \((\nabla \theta _{=k})_{1 \le k \le m}\), the \(\sigma \)-algebras of \(\nabla \theta _{\ell -1 \le \cdot \le \ell }\) are independent for \(1 \le \ell \le m\), where \(\nabla \theta _{\ell -1 \le \cdot \le \ell }\) is the collection of gradients \(\theta _u - \theta _v\) with \(2^{\ell -1} \le \Vert u\Vert _1,\Vert v\Vert _1 \le 2^\ell \).
 
4
It suffices to show that \(\iint \prod _{i,j=1}^n h(s_i,t_j)d\lambda (s_i)d\lambda (t_j)>0\) for \(n\ge 1\). Fubini’s theorem reduces this to \(\iint \prod _{i=1}^n h(s_i,t)d\lambda (s_i)d\lambda (t)>0\), which then follows from Fubini’s theorem and the assumption on h.
 
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Metadata
Title
Lectures on the Spin and Loop O(n) Models
Authors
Ron Peled
Yinon Spinka
Copyright Year
2019
Publisher
Springer Singapore
DOI
https://doi.org/10.1007/978-981-15-0294-1_10