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

Abstract

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 SAWs hitting the side of the rectangle to the fraction that hit the end of the rectangle can be calculated. By considering rectangles of fixed aspect ratio 2, and also rectangles of aspect ratio 10, we calculate this ratio exactly for larger and larger rectangles. By extrapolating this data to infinite rectangle size, and invoking the above conjectures, we obtain the estimate κ = 2.666 64 ± 0.000 07 for rectangles of aspect ratio 2 and κ = 2.666 75 ± 0.000 15 for rectangles of aspect ratio 10. We also provide numerical evidence supporting the conjectured distribution of SAWs striking the boundary at various points in the case of rectangles with aspect ratio 2.