Supercritical self-avoiding walks are space-filling

Hugo Duminil-Copin; Gady Kozma; Ariel Yadin

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 2, page 315-326
  • ISSN: 0246-0203

Abstract

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In this article, we consider the following model of self-avoiding walk: the probability of a self-avoiding trajectory γ between two points on the boundary of a finite subdomain of ℤ d is proportional to μ - length ( γ ) . When μ is supercritical (i.e. μ l t ; μ c where μ c is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.

How to cite

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Duminil-Copin, Hugo, Kozma, Gady, and Yadin, Ariel. "Supercritical self-avoiding walks are space-filling." Annales de l'I.H.P. Probabilités et statistiques 50.2 (2014): 315-326. <http://eudml.org/doc/272072>.

@article{Duminil2014,
abstract = {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 $\mu ^\{-\mbox\{length\}(\gamma )\}$. When $\mu $ is supercritical (i.e. $\mu &lt;\mu _\{c\}$ where $\mu _\{c\}$ is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.},
author = {Duminil-Copin, Hugo, Kozma, Gady, Yadin, Ariel},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {self avoiding walk; connective constant},
language = {eng},
number = {2},
pages = {315-326},
publisher = {Gauthier-Villars},
title = {Supercritical self-avoiding walks are space-filling},
url = {http://eudml.org/doc/272072},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Duminil-Copin, Hugo
AU - Kozma, Gady
AU - Yadin, Ariel
TI - Supercritical self-avoiding walks are space-filling
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 2
SP - 315
EP - 326
AB - 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 $\mu ^{-\mbox{length}(\gamma )}$. When $\mu $ is supercritical (i.e. $\mu &lt;\mu _{c}$ where $\mu _{c}$ is the connective constant of the lattice), we show that the random trajectory becomes space-filling when taking the scaling limit.
LA - eng
KW - self avoiding walk; connective constant
UR - http://eudml.org/doc/272072
ER -

References

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  1. [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. Zbl1317.60125MR3025395
  2. [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. Zbl1246.82042
  3. [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. Zbl1193.82014MR2525670
  4. [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. Zbl1230.82028MR2827969
  5. [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. Zbl0575.60099MR782962
  6. [6] H. Duminil-Copin and A. Hammond. Self-avoiding walk is sub-ballistic. Preprint. Available at arXiv:1205.0401. Zbl1277.82027MR3117515
  7. [7] H. Duminil-Copin and S. Smirnov. The connective constant of the honeycomb lattice equals 2 + 2 . Ann. of Math. (2) 175(3) (2012) 1653–1665. Zbl1253.82012MR2912714
  8. [8] P. Flory. Principles of Polymer Chemistry. Cornell University Press, Ithaca, 1953. 
  9. [9] G. Grimmett. Percolation. Springer, Berlin, 1999. Zbl0926.60004MR1707339
  10. [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. Zbl0123.00304MR139535
  11. [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. Zbl0728.60103MR1093059
  12. [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. Zbl0755.60053MR1171762
  13. [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. Zbl46.0262.03JFM46.0262.03
  14. [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. Zbl0924.60086MR1670027
  15. [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. Zbl1069.60089MR2112127
  16. [16] N. Madras and G. Slade. The Self-Avoiding Walk. Probability and Its Applications. Birkhäuser, Boston, MA, 1993. Zbl0872.60076MR1197356
  17. [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. Zbl1112.82014MR2275653

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