flatIGW - an inverse algorithm to compute the Density of States of lattice Self Avoiding Walks

TL;DR

This paper introduces flatIGW, an inverse algorithm that dynamically adjusts step-growth rules to accurately estimate the Density of States for lattice Self Avoiding Walks, enabling precise determination of the $ heta$ temperature.

Contribution

The paper presents a novel inverse algorithm, flatIGW, that effectively estimates the DoS for lattice Self Avoiding Walks by using a flat energy histogram to guide walk growth.

Findings

Successfully estimated the DoS for various walk lengths.

Located the $ heta$ temperature close to the known value.

Produced complex Fisher zeros consistent with theoretical expectations.

Abstract

We show that the Density of States (DoS) for lattice Self Avoiding Walks can be estimated by using an inverse algorithm, called flatIGW, whose step-growth rules are dynamically adjusted by requiring the energy histogram to be locally flat. Here, the (attractive) energy associated with a configuration is taken to be proportional to the number of non-bonded nearest neighbor pairs (contacts). The energy histogram is able to explicitly direct the growth of a walk because the step-growth rule of the Interacting Growth Walk \cite{IGW} samples the available nearest neighbor sites according to the number of contacts they would make. We have obtained the complex Fisher zeros corresponding to the DoS, estimated for square lattice walks of various lengths, and located the $\theta$ temperature by extrapolating the finite size values of the real zeros to their asymptotic value, $\sim 1.49$ (reasonably close to the known value, $\sim 1.50$ \cite{barkema}).