Journal article

A series test of the scaling limit of self-avoiding walks

Anthony J Guttmann, Jesper L Jacobsen



It is widely believed that the scaling limit of self-avoiding walks (SAWs) at the critical temperature is conformally invariant, and consequently describable by Schramm-Loewner evolution with parameter κ = 8/3. We consider SAWs in a rectangle, which originate at its centre and end at the boundary. We assume that the boundary density transforms covariantly in a way that depends precisely on κ, as conjectured by Lawler, Schramm and Werner (2004 Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot part 2, pp 339-64). It has previously been shown by Guttmann and Kennedy (2013 J. Eng. Math. at press) that, in the limit of an infinitely large rectangle, the ratio of the fraction of SA..

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University of Melbourne Researchers


Awarded by Australian Research Council

Awarded by Agence Nationale de la Recherche

Funding Acknowledgements

This work was supported by the Australian Research Council through grant DP120100939 (AJG and JLJ), and by the Institut Universitaire de France and Agence Nationale de la Recherche through grant ANR-10-BLAN-0414 (JLJ). AJG would like to thank Larry Glasser for a lesson in asymptotic expansions, Tom Kennedy for a lesson in conformal mapping, and both authors would like to thank Hugo Duminil-Copin, Tom Kennedy and the anonymous referees for helpful comments that improved the current manuscript, and to gratefully acknowledge the support of the Simons Center for Geometry and Physics at SUNY Stony Brook where this work was, in part, carried out.