Efficient Simulation of Self-Avoiding Walks*

Self-avoiding walks are simply defined as walks on a lattice that avoid themselves, and provide the simplest model of a polymer that captures universal features such as the Flory exponent ν which characterises the size of a polymer. In recent years, high precision Monte Carlo simulations of self-avoiding walks with many millions of steps have been realised through the use of a radically efficient implementation of the pivot algorithm via a hierarchical data structure. This data structure allows for global updates of the system to be performed in the same CPU time as local updates.

I will describe the key geometric intuition behind this implementation, and outline its application to the calculation of various quantities for self-avoiding walks, such as the critical exponents ν = 0.58759700(40) [1] and γ = 1.156 953 00(95) [2] for three-dimensional walks, and the study of logarithmic corrections for four-dimensional walks [3].

Finally, I will discuss some recent extensions of the method to dense polymer systems and to continuum models of polymers, and will speculate on possible future applications of fast global Monte Carlo moves to other models in statistical physics.

[1] Nathan Clisby and Burkhard Dünweg, High precision estimate of the hydrodynamic radius for self-avoiding walks, Phys. Rev. E. 94: 052102 (2016).
[2] Nathan Clisby, Scale-free Monte Carlo method for calculating the critical exponent γ of self-avoiding walks, J. Phys. A.: Math. Theor. 50: 264003 (2017).
[3] Nathan Clisby, Monte Carlo study of four-dimensional self-avoiding walks of up to one billion steps, J. Stat. Phys. 172:477–49 (2018).