Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Interacting self-avoiding polygons

Volker Betz, Helge Schäfer, and Lorenzo Taggi

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Abstract

We consider a system of self-avoiding polygons interacting through a potential that penalizes or rewards the number of mutual touchings and we provide an exact computation of the critical curve separating a regime of long polygons from a regime of localized polygons. Moreover, we prove the existence of a sub-region of the phase diagram where the self-avoiding polygons are space filling and we provide a non-trivial characterization of the regime where the polygon length admits uniformly bounded exponential moments.

Résumé

Dans cet article, nous considérons un système de polygones auto-évitants interagissant au moyen d’un potentiel qui pénalise ou récompense le nombre de points de contacts. Nous calculons la forme exacte de la courbe critique séparant un régime de polygones longs d’un régime de polygones localisés. En outre, nous prouvons l’existence d’un sous-domaine du diagramme de phase dans lequel les polygones remplissent l’espace dans un sens faible et donnons une caractérisation non-triviale du sous-régime où les longueurs des polygones ont des moments exponentiels bornés.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1321-1335.

Dates
Received: 15 October 2018
Revised: 18 March 2019
Accepted: 7 May 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345639

Digital Object Identifier
doi:10.1214/19-AIHP1003

Mathematical Reviews number (MathSciNet)
MR4076785

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B27: Critical phenomena 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Keywords
Phase transition Weakly space filling loops Spatial random permutations

Citation

Betz, Volker; Schäfer, Helge; Taggi, Lorenzo. Interacting self-avoiding polygons. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1321--1335. doi:10.1214/19-AIHP1003. https://projecteuclid.org/euclid.aihp/1584345639


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References

  • [1] V. Betz and L. Taggi. Scaling limit of a self-avoiding walk interacting with spatial random permutations, 2017. Available at arXiv:1612.07234.
  • [2] V. Betz and D. Ueltschi. Spatial random permutations and infinite cycles. Comm. Math. Phys. 285 (2009) 469–501.
  • [3] V. Betz and D. Ueltschi. Critical temperature of dilute Bose gases. Phys. Rev. A 81 (2010) 023611.
  • [4] F. den Hollander. Random Polymers. École d’Été de Probabilités de Saint-Flour XXXVII – 2007, Lecture Notes in Mathematics. Springer, ISSN. print edition: 0075-8434.
  • [5] H. Duminil-Copin, G. Kozma and A. Yadin. Supercritical self-avoiding walks are space-filling. Ann. Inst. Henri Poincaré B, Probab. Stat. 50 (2) (2014) 315–326.
  • [6] R. P. Feynman. Atomic theory of the $\lambda $ transition in helium. Phys. Rev. 91 (1953) 1291–1301.
  • [7] R. Kenyon. Conformal invariance of loops in the double dimer model. Comm. Math. Phys. 2 (2014).
  • [8] R. Kikuchi. $\lambda $ transition of liquid helium. Phys. Rev. 96 (1954) 563–568.
  • [9] N. Madras and G. Slade. The Self-Avoiding Walk. Birkhäuser, 2013. Reprint of the 1996 Edition.