Supercritical self-avoiding walks are space-filling

Duminil-Copin, Hugo; Kozma, Gady; Yadin, Ariel

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 315-326.

Dans cet article, nous considérons le modèle suivant de marches auto-évitantes : la probabilité d’une trajectoire auto-évitante $\gamma$ entre deux points fixés d’un sous-domaine fini de ${\mathbb{Z}}^d$ est proportionnelle à ${\mu }^{-\text{length}\left(\gamma \right)}$. Lorsque le paramètre $\mu$ est supercritique (i.e. $\mu <{\mu }_c$ ou ${\mu }_c$ est la constante de connectivité du réseau), nous prouvons que la trajectoire aléatoire remplit l’espace lorsque l’on considère la limite d’échelle du modèle.

In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory $\gamma$ between two points on the boundary of a finite subdomain of ${\mathbb{Z}}^d$ is proportional to ${\mu }^{-\text{length}\left(\gamma \right)}$. When $\mu$ is supercritical (i.e. $\mu <{\mu }_c$ where ${\mu }_c$ is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.

[1] R. Bauerschmidt, H. Duminil-Copin, J. Goodman and G. Slade. Lectures on self-avoiding-walks. In Probability and Statistical Physics in Two and More Dimensions 395-467. D. Ellwood, C. Newman, V. Sidoravicius and W. Werner (Eds). Clay Math. Proc. 15. Amer. Math. Soc., Providence, RI, 2012. Available at arXiv:1109.1549. | MR

[2] D. C. Brydges, A. Dahlqvist and G. Slade. The strong interaction limit of continuous-time weakly self-avoiding walk. Preprint. Available at arXiv:1104.3731. | Zbl

[3] D. C. Brydges, J. Z. Imbrie and G. Slade. Functional integral representations for self-avoiding walk. Probab. Surv. 6 (2009) 34-61. Available at i-journals.org. | MR | Zbl

[4] D. C. Brydges and G. Slade. Renormalisation group analysis of weakly self-avoiding walk in dimensions four and higher. In Proceedings of the International Congress of Mathematicians 2232-2257. Hindustian Book Agency, New Delhi. Available at arXiv:1003.4484. | MR | Zbl

[5] D. C. Brydges and T. Spencer. Self-avoiding walk in $5$ or more dimensions. Comm. Math. Phys. 97(1-2) (1985) 125-148. Available at projecteuclid.org. | MR | Zbl

[6] H. Duminil-Copin and A. Hammond. Self-avoiding walk is sub-ballistic. Preprint. Available at arXiv:1205.0401. | MR | Zbl

[7] H. Duminil-Copin and S. Smirnov. The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$. Ann. of Math. (2) 175(3) (2012) 1653-1665. | MR | Zbl

[8] P. Flory. Principles of Polymer Chemistry. Cornell University Press, Ithaca, 1953.

[9] G. Grimmett. Percolation. Springer, Berlin, 1999. | MR | Zbl

[10] J. M. Hammersley and D. J. A. Welsh. Further results on the rate of convergence to the connective constant of the hypercubical lattice. Quart. J. Math. Oxford Ser. (2) 13 (1962) 108-110. Available at oxfordjournals.org. | MR | Zbl

[11] T. Hara and G. Slade. Critical behaviour of self-avoiding walk in five or more dimensions. Bull. Amer. Math. Soc. (N.S.) 25(2) (1991) 417-423. Available at ams.org. | MR | Zbl

[12] T. Hara and G. Slade. Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys. 147(1) (1992) 101-136. Available at projecteuclid.org. | MR | Zbl

[13] G. H. Hardy and S. Ramanujan. The normal number of prime factors of a number $n$. Quart. J. Pure Appl. Math. 48 (1917) 76-92. | JFM

[14] D. Ioffe. Ornstein-Zernike behaviour and analyticity of shapes for self-avoiding walks on ${𝐙}^d$. Markov Process. Related Fields 4(3) (1998) 323-350. | MR | Zbl

[15] G. F. Lawler, O. Schramm and W. Werner. On the scaling limit of planar self-avoiding walk. In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2 339-364. Proc. Sympos. Pure Math. 72. Amer. Math. Soc., Providence, RI, 2004. Available at arXiv:math/0204277. | MR | Zbl

[16] N. Madras and G. Slade. The Self-Avoiding Walk. Probability and Its Applications. Birkhäuser, Boston, MA, 1993. | MR | Zbl

[17] S. Smirnov. Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians, Vol. II 1421-1451. Eur. Math. Soc., Zürich, 2006. Available at arXiv:0708.0032. | MR | Zbl