Electronic Communications in Probability

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

Lorenzo Taggi

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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

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

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

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Zentralblatt MATH identifier

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]

loop O(n) model loop soups phase transition

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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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  • [1] V. Betz, L. Taggi: Scaling limit of a self-avoiding walk interacting with random spatial permutations, arXiv:1612.07234 (2017).
  • [2] V. Betz, H. Schäfer, L. Taggi: Interacting self-avoiding polygons, arXiv:1805.08517 (2018).
  • [3] L. Chayes, L. P. Pryadko, K. Shtengel: Intersecting loop models on $\mathbb{Z} ^d$: rigorous results, Nuclear Physics B, 570, 590–641 (2000).
  • [4] E. Domany, D Mukamel, B. Nienhuis, and A Schwimmer: Duality relations and equivalences for models with O(n) and cubic symmetry, Nuclear Physics B, 190(2):279–287, (1981).
  • [5] H. Duminil-Copin, G. Kozma; A. Yadin: Supercritical self-avoiding walks are space-filling, Annales de l’I.H.P. Probabilités et statistiques, Tome 50 no. 2, p. 315–326 (2014).
  • [6] H. Duminil-Copin, R. Peled W. Samotij, Y. Spinka: Exponential Decay of Loop Lengths in the Loop $O(n)$ Model with Large $n$, Commun. Math. Phys. (2017) 349: 777, https://doi.org/10.1007/s00220-016-2815-4.
  • [7] H. Duminil-Copin, A. Glazman, R. Peled, Y. Spinka: Macroscopic loops in the loop $O(n)$ model at Nienhuis’ critical point, arXiv:1707.09335 (2017).
  • [8] H. Duminil-Copin and S. Smirnov: The connective constant of the honeycomb lattice equals $\sqrt{2 + \sqrt {2}} $, Annals of Mathematics, Second Series, Vol. 175, No. 3 (May, 2012), pp. 1653–1665
  • [9] R. P. Feynman: Atomic Theory of the $\lambda $ Transition in Helium, Phys. Rev. 91, 1291 – (1953).
  • [10] S. Grosskinsky, A. A. Lovisolo, D. Ueltschi: Lattice permutations and Poisson-Dirichlet distribution of cycle lengths, J. Statist. Phys. 146, 1105–1121 (2012).
  • [11] J. M. Hammersley and K. M. Morton: Poor man’s Monte Carlo, J. R. Statist. Soc. B 16 23 (1954).
  • [12] J. M. Hammersley: The number of polygons on a lattice, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 57, Issue 3, pp. 516–523 (1961).
  • [13] J. M. Hammersley, S. Whittington: Self-avoiding walks in wedges, J. Phys. A: Math. Gen. 18 101–111 (1985).
  • [14] F. den Hollander, Random Polymers, École d’Été Probabilités de Saint-Flour XXXVII – 2007, Springer DOI: 10.1007/978-3-642-00333-2 (2009).
  • [15] H. Kesten: On the number of self-avoiding walks, J. Math. Phys. 4 (1963) 960–969.
  • [16] N. Madras, G. Slade: The self-avoiding walk, Birkhäuser, preprint of the 1996 Edition.
  • [17] B. Nienhuis: Exact Critical Point and Critical Exponents of $O(n)$ Models in Two Dimensions, Phys. Rev. Lett. 49, 1062 (1982).
  • [18] B. Nienhuis: Coulomb gas description of 2D critical behaviour. J. Statist. Phys., 34:731–761, (1984).
  • [19] B. Nienhuis: Locus of the tricritical transition in a two-dimensional q-state potts model. Physica A: Statistical Mechanics and its Applications, 177(1–3):109–113, (1991).
  • [20] R. Peled, Y. Spinka: Lectures on the spin and Loop $O(n)$ models, arXiv:1708.00058 (2017).