The Annals of Probability

On the probability that self-avoiding walk ends at a given point

Hugo Duminil-Copin, Alexander Glazman, Alan Hammond, and Ioan Manolescu

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We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on $\mathbb{Z}^{d}$ for $d\geq2$. We show that the probability that a walk of length $n$ ends at a point $x$ tends to $0$ as $n$ tends to infinity, uniformly in $x$. Also, when $x$ is fixed, with $\Vert x\Vert=1$, this probability decreases faster than $n^{-1/4+\varepsilon}$ for any $\varepsilon>0$. This provides a bound on the probability that a self-avoiding walk is a polygon.

Article information

Ann. Probab., Volume 44, Number 2 (2016), 955-983.

Received: May 2013
Revised: September 2014
First available in Project Euclid: 14 March 2016

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Self-avoiding walk self-avoiding polygons endpoint delocalization


Duminil-Copin, Hugo; Glazman, Alexander; Hammond, Alan; Manolescu, Ioan. On the probability that self-avoiding walk ends at a given point. Ann. Probab. 44 (2016), no. 2, 955--983. doi:10.1214/14-AOP993.

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  • [1] Bauerschmidt, R., Duminil-Copin, H., Goodman, J. and Slade, G. (2012). Lectures on self-avoiding walks. In Probability and Statistical Physics in Two and More Dimensions (D. Ellwood, C. Newman, V. Sidoravicius and W. Werner, eds.). Clay Math. Proc. 15 395–467. Amer. Math. Soc., Providence, RI.
  • [2] Beaton, N. R., Bousquet-Mélou, M., de Gier, J., Duminil-Copin, H. and Guttmann, A. J. (2014). The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is $1+\sqrt{2}$. Comm. Math. Phys. 326 727–754.
  • [3] Brydges, D. and Slade, G. (2010). Renormalisation group analysis of weakly self-avoiding walk in dimensions four and higher. In Proceedings of the International Congress of Mathematicians. Volume IV 2232–2257. Hindustan Book Agency, New Delhi.
  • [4] Brydges, D. C., Imbrie, J. Z. and Slade, G. (2009). Functional integral representations for self-avoiding walk. Probab. Surv. 6 34–61.
  • [5] Clisby, N. and Slade, G. (2009). Polygons and the lace expansion. In Polygons, Polyominoes and Polycubes. Lecture Notes in Physics 775 117–142. Springer, Dordrecht.
  • [6] Duminil-Copin, H. and Hammond, A. (2013). Self-avoiding walk is sub-ballistic. Comm. Math. Phys. 324 401–423.
  • [7] Duminil-Copin, H. and Smirnov, S. (2012). The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$. Ann. of Math. (2) 175 1653–1665.
  • [8] Duplantier, B. (1989). Fractals in two dimensions and conformal invariance. Fractals in physics (Vence, 1989). Phys. D 38 71–87.
  • [9] Duplantier, B. (1990). Renormalization and conformal invariance for polymers. In Fundamental Problems in Statistical Mechanics VII (Altenberg, 1989) 171–223. North-Holland, Amsterdam.
  • [10] Flory, P. (1953). Principles of Polymer Chemistry. Cornell Univ. Press, Ithaca, NY.
  • [11] Glazman, A. (2014). Connective constant for a weighted self-avoiding walk on $\mathbb{Z}^{2}$. Preprint. Available at arXiv:1402.5376.
  • [12] Hammersley, J. M. and Welsh, D. J. A. (1962). Further results on the rate of convergence to the connective constant of the hypercubical lattice. Quart. J. Math. Oxford Ser. (2) 13 108–110.
  • [13] Hara, T. and Slade, G. (1991). Critical behaviour of self-avoiding walk in five or more dimensions. Bull. Amer. Math. Soc. (N.S.) 25 417–423.
  • [14] Hara, T. and Slade, G. (1992). Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys. 147 101–136.
  • [15] Janse van Rensburg, E. J., Orlandini, E., Sumners, D. W., Tesi, M. C. and Whittington, S. G. (1993). The writhe of a self-avoiding polygon. J. Phys. A 26 L981–L986.
  • [16] Kesten, H. (1963). On the number of self-avoiding walks. J. Math. Phys. 4 960–969.
  • [17] Lawler, G. F., Schramm, O. and Werner, W. (2004). On the scaling limit of planar self-avoiding walk. In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2. Proc. Sympos. Pure Math. 72 339–364. Amer. Math. Soc., Providence, RI.
  • [18] Madras, N. (1995). A rigorous bound on the critical exponent for the number of lattice trees, animals, and polygons. J. Stat. Phys. 78 681–699.
  • [19] Madras, N. (2014). A lower bound for the end-to-end distance of the self-avoiding walk. Canad. Math. Bull. 57 113–118.
  • [20] Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston, MA.
  • [21] Nienhuis, B. (1982). Exact critical point and critical exponents of $\mathrm{O}(n)$ models in two dimensions. Phys. Rev. Lett. 49 1062–1065.
  • [22] Nienhuis, B. (1984). Coulomb gas description of 2D critical behaviour. J. Stat. Phys. 34 731–761.
  • [23] Orr, W. J. C. (1947). Statistical treatment of polymer solutions at infinite dilution. Trans. Faraday Soc. 43 12–27.