The lace expansion was introduced by Brydges and Spencer in 1985 [7] to analyse weakly self-avoiding walks in dimensions d > 4. Subsequently it has been generalised and greatly extended, so that it now applies to a variety of problems of interest in probability theory, statistical physics, and combinatorics, including the strictly self-avoiding walk, lattice trees, lattice animals, percolation, oriented percolation, the contact process, random graphs, and the Ising model. A recent survey is [42].
In this chapter, we give an introduction to the lace expansion for self-avoiding walks, with emphasis on self-avoiding polygons. We focus on combinatorial rather than analytical aspects.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
D. Aldous. Tree-based models for random distribution of mass. J. Stat. Phys., 73:625–641, (1993).
A. Bovier, J. Fröhlich, and U. Glaus. Branched polymers and dimensional reduction. In K. Osterwalder and R. Stora, editors, Critical Phenomena, Random Systems, Gauge Theories, Amsterdam, (1986). North-Holland.
D.C. Brydges and J.Z. Imbrie. Branched polymers and dimensional reduction. Ann. Math., 158:1019–1039, (2003).
D.C. Brydges and J.Z. Imbrie. Dimensional reduction formulas for branched polymer correlation functions. J. Stat. Phys., 110:503–518, (2003).
D.C. Brydges and J.Z. Imbrie. End to end distance from the Green's function for a hierarchical self-avoiding walk in four dimensions. Commun. Math. Phys., 239:523–1547, (2003).
D.C. Brydges and J.Z. Imbrie. Green's function for a hierarchical self-avoiding walk in four dimensions. Commun. Math. Phys., 239:549–584, (2003).
D.C. Brydges and T. Spencer. Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys., 97:125–148, (1985).
N. Clisby, R. Liang, and G. Slade. Self-avoiding walk enumeration via the lace expansion. J. Phys. A: Math. Theor., 40:10973–11017, (2007).
E. Derbez and G. Slade. The scaling limit of lattice trees in high dimensions. Commun. Math. Phys., 193:69–104, (1998).
M.E. Fisher and D.S. Gaunt. Ising model and self-avoiding walks on hypercubical lattices and “high-density” expansions. Phys. Rev., 133:A224–A239, (1964).
M.E. Fisher and R.R.P. Singh. Critical points, large-dimensionality expansions, and the Ising spin glass. In G.R. Grimmett and D.J.A. Welsh, editors, Disorder in Physical Systems. Clarendon Press, Oxford, (1990).
D.S. Gaunt. 1/d expansions for critical amplitudes. J. Phys. A: Math. Gen., 19:L149–L153, (1986).
P.R. Gerber and M.E. Fisher. Critical temperatures of classical n-vector models on hypercubic lattices. Phys. Rev. B, 10:4697–4703, (1974).
A.J. Guttmann and S.G. Whittington. Two-dimensional lattice embeddings of connected graphs of cyclomatic index two. J. Phys. A: Math. Gen., 11:721–729, (1978).
A.J. Guttmann. Correction to scaling exponents and critical properties of the n-vector model with dimensionality > 4. J. Phys. A: Math. Gen., 14:233–239, (1981).
A.J. Guttmann. Asymptotic analysis of power-series expansions. In C. Domb and J.L. Lebowitz, editors, Phase Transitions and Critical Phenomena, Volume 13, pages 1–234. Academic Press, New York, (1989).
J.M. Hammersley. The number of polygons on a lattice. Proc. Camb. Phil. Soc., 57:516–523, (1961).
T. Hara and G. Slade. On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys., 59:1469–1510, (1990).
T. Hara and G. Slade. The lace expansion for self-avoiding walk in five or more dimensions. Rev. Math. Phys., 4:235–327, (1992).
T. Hara and G. Slade. The number and size of branched polymers in high dimensions. J. Stat. Phys., 67:1009–1038, (1992).
T. Hara and G. Slade. Self-avoiding walk in five or more dimensions. I. The critical behaviour. Commun. Math. Phys., 147:101–136, (1992).
T. Hara and G. Slade. The self-avoiding-walk and percolation critical points in high dimensions. Combin. Probab. Comput., 4:197–215, (1995).
R. van der Hofstad and G. Slade. A generalised inductive approach to the lace expansion. Probab. Theory Related Fields, 122:389–430, (2002).
R. van der Hofstad and G. Slade. The lace expansion on a tree with application to networks of self-avoiding walks. Adv. Appl. Math., 30:471–528, (2003).
M. Holmes. Convergence of lattice trees to super-Brownian motion above the critical dimension. Electr. J. Prob., 13:671–755, (2008).
M. Holmes, A.A. Járai, A. Sakai, and G. Slade. High-dimensional graphical networks of self-avoiding walks. Canad. J. Math., 56:77–114, (2004).
E.J. Janse van Rensburg. On the number of trees in f d. J. Phys. A: Math. Gen., 25:3523– 3528, (1992).
I. Jensen. Enumeration of self-avoiding walks on the square lattice. J. Phys. A: Math. Gen., 37:5503–5524, (2004).
I. Jensen. Self-avoiding walks and polygons on the triangular lattice. J. Stat. Mech., P10008, (2004).
I. Jensen. Honeycomb lattice polygons and walks as a test of series analysis techniques. J. Phys.: Conf. Series, 42:163–178, (2006).
R. Kenyon and P. Winkler. Branched polymers in 2 and 3 dimensions. To appear in Amer. Math. Monthly.
H. Kesten. On the number of self-avoiding walks. J. Math. Phys., 4:960–969, (1963).
D.J. Klein. Rigorous results for branched polymer models with excluded volume. J. Chem. Phys., 75:5186–5189, (1981).
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, pages 339–364, Proc. Sympos. Pure Math., 72, Part 2. Amer. Math. Soc., Providence, RI, (2004).
T.C. Lubensky and J. Isaacson. Statistics of lattice animals and dilute branched polymers. Phys. Rev., A20:2130–2146, (1979).
N. Madras. Bounds on the critical exponent of self-avoiding polygons. In R. Durrett and H. Kesten, editors, Random Walks, Brownian Motion and Interacting Particle Systems, Boston, (1991). Birkhaüser.
N. Madras. A rigorous bound on the critical exponent for the number of lattice trees, animals and polygons. J. Stat. Phys., 78:681–699, (1995).
A.M. Nemirovsky, K.F. Freed, T. Ishinabe, and J.F. Douglas. End-to-end distance of a single self-interacting self-avoiding polymer chain: d −1 expansion. Phys. Lett. A, 162:469–474, (1992).
A.M. Nemirovsky, K.F. Freed, T. Ishinabe, and J.F. Douglas. Marriage of exact enumeration and 1/d expansion methods: Lattice model of dilute polymers. J. Stat. Phys., 67:1083–1108, (1992).
G. Parisi and N. Sourlas. Critical behavior of branched polymers and the Lee—Yang edge singularity. Phys. Rev. Lett., 46:871–874, (1981).
E. Perkins. Super-Brownian motion and critical spatial stochastic systems. Canad. Math. Bull., 47:280–297, (2004).
G. Slade. The Lace Expansion and its Applications. Springer, Berlin, (2006). Lecture Notes in Mathematics Vol. 1879. Ecole d'Eté de Probabilités de Saint—Flour XXXIV—2004.
H. Tasaki. Stochastic geometric methods in Statistical Physics and Field Theories. PhD thesis, University of Tokyo, (1986).
H. Tasaki and T. Hara. Critical behaviour in a system of branched polymers. Prog. Theor. Phys. Suppl., 92:14–25, (1987).
M.J. Velgakis, G.A. Baker, Jr., and J. Oitmaa. Integral approximants. Comput. Phys. Commun., 99:307–322, (1997).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Canopus Academic Publishing Limited
About this chapter
Cite this chapter
Clisby, N., Slade, G. (2009). Polygons and the Lace Expansion. In: Guttman, A.J. (eds) Polygons, Polyominoes and Polycubes. Lecture Notes in Physics, vol 775. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9927-4_6
Download citation
DOI: https://doi.org/10.1007/978-1-4020-9927-4_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-9926-7
Online ISBN: 978-1-4020-9927-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)