Self-avoiding walks contained within a square

Abstract

AbstractWe have studied self-avoiding walks contained within anL×Lsquare whose end-points can lie anywhere within, or on, the boundaries of the square. We prove that such walks behave, asymptotically, as walks crossing a square (WCAS), being those walks whose end-points lie at the south-east and north-west corners of the square. We provide numerical data, enumerating all such walks, and analyse the sequence of coefficients in order to estimate the asymptotic behaviour. We also studied a subset of these walks, those that must contain at least one edge on all four boundaries of the square. We provide compelling evidence that these two classes of walks grow identically. From our analysis we conjecture that the number of such walksCL, for both problems, behaves asCL∼λL2+bL+c⋅Lg,where (Guttmann and Jensen 2022J. Phys. A: Math. Theor.)λ= 1.744 5498 ± 0.000 0012,b= −0.043 54 ± 0.0005,c= −1.35 ± 0.45, andg= 3.9 ± 0.1. Finally, we also studied the equivalent problem for self-avoiding polygons, also known as cycles in a square grid. The asymptotic behaviour of cycles has the same form as walks, but with different values of the parametersc, andg. Our numerical analysis shows thatλandbhave the same values as for WCAS and thatc= 1.776 ± 0.002 whileg= −0.500 ± 0.005 and hence probably equals−12.