Qiuyi Li is a bachelor student in the Department of Mathematics and Statistics (with a specialisation in applied mathematics) at The University of Melbourne. He is interested in discrete random walks. More specifically, his work examines the asymptotic behaviour of the number of eSAWs of length n and the probability (in relation to length) that a randomly generated SAW is endless.
Endless Self-Avoiding Walks in Two Dimensions
In this project, we will focus on a new variant of SAW, endless self-avoiding walks (eSAW) in two- dimensional lattices. An endless self-avoiding walk is a self-avoiding walk which, when concatenated with copies of itself head-to-tail ad infinitum, remains self-avoiding, which first introduced by Nathan Clisby.
We will first investigate the asymptotic behavior of e_n, the number of eSAWs of length n, using theoretical analysis. Then we will compute the probability (in relation to length) that a randomly generated SAW is endless, using the well-known pivot algorithm to simulate SAWs. By combining this with the known asymptotic growth of c_n, then number of self-avoiding walks of length n, we can then deduce the asymptotic growth of e_n.