Electronic Communications in Probability

Shifted critical threshold in the loop $ \boldsymbol{O(n)} $ model at arbitrarily small $n$

Lorenzo Taggi

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Abstract

In the loop $O(n)$ model a collection of mutually-disjoint self-avoiding loops is drawn at random on a finite domain of a lattice with probability proportional to \[\lambda ^{ \# \mbox{ edges} } n^{ \# \mbox{ loops} },\] where $\lambda , n \in [0, \infty )$. Let $\mu $ be the connective constant of the lattice and, for any $n \in [0, \infty )$, let $\lambda _c(n)$ be the largest value of $\lambda $ such that the loop length admits uniformly bounded exponential moments. It is not difficult to prove that $\lambda _c(n) =1/\mu $ when $n=0$ (in this case the model corresponds to the self-avoiding walk) and that for any $n \geq 0$, $\lambda _c(n) \geq 1/\mu $. In this note we prove that, \[ \lambda _c(n) > 1/\mu \, \, \, \, \forall n >0, \] \[\lambda _c(n) \geq 1/\mu \, + \, c_0 \, n \, + \, O(n^2) \, \, \mbox{ as $n \rightarrow 0$,} \] on $\mathbb{Z} ^d$, with $d \geq 2$, and on the hexagonal lattice, where $c_0>0$. This means that, when $n$ is positive (even arbitrarily small), as a consequence of the mutual repulsion between the loops, a phase transition can only occur at a strictly larger critical threshold than in the self-avoiding walk.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 96, 9 pp.

Dates
Received: 16 July 2018
Accepted: 6 November 2018
First available in Project Euclid: 18 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1545102492

Digital Object Identifier
doi:10.1214/18-ECP189

Mathematical Reviews number (MathSciNet)
MR3896834

Zentralblatt MATH identifier
07023482

Subjects
Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 2B26
Secondary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
loop O(n) model loop soups phase transition

Rights
Creative Commons Attribution 4.0 International License.

Citation

Taggi, Lorenzo. Shifted critical threshold in the loop $ \boldsymbol{O(n)} $ model at arbitrarily small $n$. Electron. Commun. Probab. 23 (2018), paper no. 96, 9 pp. doi:10.1214/18-ECP189. https://projecteuclid.org/euclid.ecp/1545102492


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