Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Interacting self-avoiding polygons

Volker Betz, Helge Schäfer, and Lorenzo Taggi

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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.


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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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

Phase transition Weakly space filling loops Spatial random permutations


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.

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