A series test of the scaling limit of self-avoiding walks (Q2857973)

It is widely believed that the scaling limit of critical self-avoiding walk (SAW) on the square lattice is conformally invariant and given by the Schramm-Loewner evolution with parameter \(\kappa = 8/3\). The aim of the present paper is to support the above conjecture by providing numerical estimates on \(\kappa\).NEWLINENEWLINEThe authors consider SAWs in rectangles starting from the center and ending at the boundary and calculate the fraction of SAWs ending at the horizontal part of the boundary for larger and larger rectangles of aspect ratios \(2\) and \(10\). An explicit asymptotic formula for such a fraction is obtained in an earlier work of \textit{A. J. Guttmann} and \textit{T. Kennedy} [``Self-avoiding walks in a rectangle'', J. Eng. Math. 80 (2013)] for infinitely large rectangles of any given aspect ratio under the assumption that the boundary density transforms covariantly in a way that depends on \(\kappa\), as conjectured by \textit{G. F. Lawler} et al. [in: Proc. Symposia Pure Math. 72, Pt. 2, 339--364 (2004; Zbl 1069.60089)]. By extrapolating their results to infinite rectangle size and comparing the outcome with the explicit formulas, the authors obtain the estimates \(\kappa = 2.66664 \pm 0.00007\) for rectangles of aspect ratio \(2\) and \(\kappa = 2.66675 \pm 0.00015\) for rectangles of aspect ratio \(10\).NEWLINENEWLINEFinally, the authors calculate the hitting distribution of the long side of a \(12\times 24\) rectangle and observe a good agreement with the conjectured hitting distribution for the infinite rectangle limit.