Abstract
We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus on the problem of reproducing the parametrization corresponding to that of lattice models, namely self-avoiding walks on the lattice, and we propose a strategy that gives rise to discrete paths where consecutive points lie an approximately constant distance apart from each other. This new method allows us to tackle two non-trivial features of self-avoiding walks that critically depend on the parametrization: the asphericity of a portion of chain and the correction-to-scaling exponent.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Des Cloizeaux, J., Jannink, G.: Polymers in Solution: Their Modelling and Structure. Oxford University Press, New York (1990)
Schäfer, L.: Excluded Volume Effects in Polymer Solutions. Springer, Berlin-New York (1999)
Madras, N., Slade, G.: The Self-avoiding Walk. Birkhäuser, Boston (1993)
Guttmann, A.J., Conway, A.R.: Square lattice self-avoiding walks and polygons. Ann. Comb. 5, 319–345 (2001)
Sokal, A.D.: Monte Carlo methods for the self-avoiding walk. Nucl. Phys. B 47, 172–179 (1996)
Janse van Rensburg, E.J.: Monte Carlo methods for the self-avoiding walk. J. Phys. A, Math. Theor. 42, 323001 (2009)
Lawler, G.F., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walks. Proc. Symp. Pure Math. 72(2), 339–364 (2004)
Kennedy, T.: The length of an SLE—Monte Carlo studies. J. Stat. Phys. 128, 1263–1277 (2007)
Gherardi, M.: Whole-plane self-avoiding walks and radial Schramm-Loewner evolution: a numerical study. J. Stat. Phys. 136, 864–874 (2009)
Lawler, G.F.: Dimension and natural parametrization for SLE curves. Preprint. arXiv:0712.3263v1 [math.PR]
Lawler, G.F., Sheffield, S.: The natural parametrization for the Schramm-Loewner evolution. Preprint. arXiv:0906.3804v1 [math.PR]
Lawler, G.F.: Conformally invariant processes in the plane. In: Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence (2005)
Werner, W.: Random planar curves and Schramm-Loewner evolutions (Lecture notes from the 2002 Saint-Flour summer school). L. N. Math. 1840, 107–195 (2004)
Cardy, J.: SLE for theoretical physicists. Ann. Phys. 318, 81–118 (2005)
Kager, W., Nienhuis, B.: A guide to stochastic Loewner evolution and its applications. J. Stat. Phys. 115, 1149–1229 (2004)
Bauer, M., Bernard, D.: 2D growth processes: SLE and Loewner chains. Phys. Rept. 432, 115–221 (2006)
Bauer, R.O.: Discrete Löwner evolution. Ann. Fac. Sci. Toulouse VI 12, 433–451 (2003)
Kennedy, T.: Monte Carlo comparisons of the self-avoiding walk and SLE as parametrized curves. Preprint. arXiv:math/0612609v2 [math.PR]
Witten, T.A., Sander, L.M.: Diffusion-limited aggregation: a kinetic critical phenomenon. Phys. Rev. Lett. 47, 1400–1403 (1981)
Hastings, M.B., Levitov, L.S.: Laplacian growth as one-dimensional turbulence. Physica D 116, 244–252 (1998)
Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)
Hastings, M.B.: Exact multifractal spectra for arbitrary Laplacian random walks. Phys. Rev. Lett. 88, 055506 (2002)
Madras, N., Sokal, A.D.: The pivot algorithm: a highly efficient Monte Carlo method for the self-avoiding walk. J. Stat. Phys. 50, 109–186 (1988)
Kennedy, T.: A Faster implementation of the pivot algorithm for self-avoiding walks. J. Stat. Phys. 106, 407–429 (2002)
Saleur, H.: Conformal invariance for polymers and percolation. J. Phys. A, Math. Gen. 20 (1987)
Caracciolo, S., Guttmann, A.J., Jensen, I., Pelissetto, A., Rogers, A.N., Sokal, A.D.: Correction-to-scaling exponents for two-dimensional self-avoiding walks. J. Stat. Phys. 120, 1037–1100 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gherardi, M. Exact Sampling of Self-avoiding Paths via Discrete Schramm-Loewner Evolution. J Stat Phys 140, 1115–1129 (2010). https://doi.org/10.1007/s10955-010-0031-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-010-0031-8