Exponential decay in the loop $O(n)$ model: $n> 1$, $x<\tfrac{1}{\sqrt{3}}+\varepsilon(n)$
| Autor: | Glazman, Alexander, Manolescu, Ioan |
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| Rok vydání: | 2018 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $\varepsilon(n)>0$. It is expected that, for $n \leq 2$, 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 $n \in (1,2]$ occurs at some critical parameter $x_c(n)$ strictly greater than that $x_c(1) = 1/\sqrt3$. 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. Comment: Compared to v1: references added, typos corrected |
| Databáze: | arXiv |
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