An upper bound on the number of self-avoiding polygons via joining. (English) Zbl 1434.60294

Summary: For \(d\geq2\) and \(n\in\mathbb{N}\) even, let \(p_{n}=p_{n}(d)\) denote the number of length \(n\) self-avoiding polygons in \(\mathbb{Z}^{d}\) up to translation. The polygon cardinality grows exponentially, and the growth rate \(\lim_{n\in2\mathbb{N}}p_{n}^{1/n}\in(0,\infty)\) is called the connective constant and denoted by \(\mu\). N. Madras [J. Stat. Phys. 78, No. 3–4, 681–699 (1995; Zbl 1080.82541)] has shown that \(p_{n}\mu^{-n}\leq Cn^{-1/2}\) in dimension \(d=2\). Here, we establish that \(p_{n}\mu^{-n}\leq n^{-3/2+o(1)}\) for a set of even \(n\) of full density when \(d=2\). We also consider a certain variant of self-avoiding walk and argue that, when \(d\geq3\), an upper bound of \(n^{-2+d^{-1}+o(1)}\) holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry


Zbl 1080.82541
Full Text: DOI arXiv


[1] Bauerschmidt, R., Brydges, D. C. and Slade, G. (2015). Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: A renormalisation group analysis. Comm. Math. Phys.337 817-877. · Zbl 1318.60049
[2] Bauerschmidt, R., Brydges, D. C. and Slade, G. (2015). A renormalisation group method. III. Perturbative analysis. J. Stat. Phys.159 492-529. · Zbl 1319.82008
[3] 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. · Zbl 1317.60125
[4] Bousquet-Mélou, M. and Brak, R. (2009). Exactly solved models. In Polygons, Polyominoes and Polycubes. Lecture Notes in Physics 775 43-78. Springer, Dordrecht. · Zbl 1226.05020
[5] Brydges, D. C. and Slade, G. (2015). A renormalisation group method. I. Gaussian integration and normed algebras. J. Stat. Phys.159 421-460. · Zbl 1317.82013
[6] Brydges, D. C. and Slade, G. (2015). A renormalisation group method. II. Approximation by local polynomials. J. Stat. Phys.159 461-491. · Zbl 1317.82014
[7] Brydges, D. C. and Slade, G. (2015). A renormalisation group method. IV. Stability analysis. J. Stat. Phys.159 530-588. · Zbl 1317.82015
[8] Brydges, D. C. and Slade, G. (2015). A renormalisation group method. V. A single renormalisation group step. J. Stat. Phys.159 589-667. · Zbl 1317.82016
[9] Duminil-Copin, H., Glazman, A., Hammond, A. and Manolescu, I. (2016). On the probability that self-avoiding walk ends at a given point. Ann. Probab.44 955-983. · Zbl 1347.60131
[10] Duplantier, B. (1989). Fractals in two dimensions and conformal invariance. Phys. D 38 71-87. Fractals in physics (Vence, 1989).
[11] Duplantier, B. (1990). Renormalization and conformal invariance for polymers. In Fundamental Problems in Statistical Mechanics VII (Altenberg, 1989) 171-223. North-Holland, Amsterdam.
[12] Flory, P. (1953). Principles of Polymer Chemistry. Cornell Univ. Press, Ithaca, NY.
[13] Hammond, A. (2017). On self-avoiding polygons and walks: Counting, joining and closing. Available at arXiv:1504.05286.
[14] Hammond, A. (2017). On self-avoiding polygons and walks: The snake method via pattern fluctuation. Available at math.berkeley.edu/alanmh/papers/snakemethodpattern.pdf. · Zbl 1478.60262
[15] Hammond, A. (2017). On self-avoiding polygons and walks: The snake method via polygon joining. Available at math.berkeley.edu/alanmh/papers/snakemethodpolygon.pdf. · Zbl 1467.60077
[16] Hara, T. and Slade, G. (1992). Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys.147 101-136. · Zbl 0755.60053
[17] Hara, T. and Slade, G. (1992). The lace expansion for self-avoiding walk in five or more dimensions. Rev. Math. Phys.4 235-327. · Zbl 0755.60054
[18] Kesten, H. (1963). On the number of self-avoiding walks. J. Math. Phys.4 960-969. · Zbl 0122.36502
[19] Lawler, G. (2013). Random walk problems motivated by statistical physics. Available at http://www.math.uchicago.edu/ lawler/russia.pdf. · Zbl 1388.60164
[20] Loomis, L. H. and Whitney, H. (1949). An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc.55 961-962. · Zbl 0035.38302
[21] Madras, N. (1991). Bounds on the critical exponent of self-avoiding polygons. In Random Walks, Brownian Motion, and Interacting Particle Systems. Progress in Probability 28 359-371. Birkhäuser, Boston, MA. · Zbl 0741.60066
[22] 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. · Zbl 1080.82541
[23] Madras, N. and Slade, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston, MA. · Zbl 0780.60103
[24] Nienhuis, B. (1982). Exact critical point and critical exponents of \(\text{O}(n)\) models in two dimensions. Phys. Rev. Lett.49 1062-1065.
[25] Nienhuis, B. (1984). Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas. J. Stat. Phys.34 731-761. · Zbl 0595.76071
[26] Orr, W. J. C. (1947). Statistical treatment of polymer solutions at infinite dilution. Trans. Faraday Soc.43 12-27.
[27] Steele, J.
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