Abstract
We show that the loop O(n) model on the hexagonal lattice exhibits exponential decay of loop sizes whenever n > 1 and \(x<\tfrac {1}{\sqrt {3}}+\varepsilon (n)\), for some suitable choice of ε(n) > 0.
It is expected that, for n ≤ 2, the model exhibits a phase transition in terms of x, that separates regimes of polynomial and exponential decay of loop sizes. In this paradigm, our result implies that the phase transition for n ∈ (1, 2] occurs at some critical parameter x c(n) strictly greater than that \(x_c(1) = 1/\sqrt {3}\). The value of the latter is known since the loop O(1) model on the hexagonal lattice represents the contours of the spin-clusters of the Ising model on the triangular lattice.
The proof is based on developing n as 1 + (n − 1) and exploiting the fact that, when \(x<\tfrac {1}{\sqrt {3}}\), the Ising model exhibits exponential decay on any (possibly non simply-connected) domain. The latter follows from the positive association of the FK-Ising representation.
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Notes
- 1.
This domination is of special interest as
and for \(x \geq 1/\sqrt {3}\). Then we may simplify the value of β as \(\beta = \frac {(2+2\sqrt {3})^6}{ (n-1)^{2} + (2+2\sqrt {3})^6} \sim 1 - \tfrac 1{(2+2\sqrt {3})^{6}} (n-1)^{2}~\). - 2.
As
and \(x \geq 1/\sqrt {3}\), we may assume that \(\alpha \sim 1 - \tfrac {1}{6\, (2+2\sqrt {3})^{6}} (n-1)^{2}\).
and for 
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Acknowledgements
The authors would like to thank Ron Peled for suggesting to develop in n = (n − 1) + 1 following Chayes and Machta. Our discussions with Hugo Duminil-Copin, Yinon Spinka and Marcelo Hilario were also very helpful. We acknowledge the hospitality of IMPA (Rio de Janeiro), where this project started.
The first author is supported by the Swiss NSF grant P300P2_177848, and partially supported by the European Research Council starting grant 678520 (LocalOrder). The second author is a member of the NCCR SwissMAP.
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Glazman, A., Manolescu, I. (2021). Exponential Decay in the Loop O(n) Model on the Hexagonal Lattice for n > 1 and \(x<\tfrac {1}{\sqrt {3}}+\varepsilon (n)\) . In: Vares, M.E., Fernández, R., Fontes, L.R., Newman, C.M. (eds) In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius. Progress in Probability, vol 77. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-60754-8_21
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