Fractality and Topology of Self-Avoiding Walks

TL;DR

This paper explores the geometric and topological properties of self-avoiding walks, introducing a new variant called loop-deleted SAW, and compares their fractal and topological features through Betti numbers and fractal dimensions.

Contribution

It introduces the loop-deleted self-avoiding walk (LDSAW), computes its critical exponent, and analyzes its fractal and topological differences from ordinary SAW.

Findings

LDSAW has a different critical exponent from SAW.

Contact-cloud exhibits multi-fractal properties.

Random subsets of SAW share the same fractal dimension as SAW.

Abstract

We have analyzed geometric and topological features of self-avoiding walks. We introduce a new kind of walk: the loop-deleted self-avoiding walk (LDSAW) motivated by the interaction of chromatin with the nuclear lamina. Its critical exponent is calculated and found to be different from that of the ordinary SAW. Taking the walks as point-clouds, the LDSAW is a subset of the SAW. We study the difference between the LDSAW and SAW by comparing their fractal dimensions and growth rates of the Betti number. In addition, the spatial distribution of the contacts inside a SAW, which is also a subset of SAW, is analyzed following the same routine. The results show that the contact-cloud has a multi-fractal property and different growth rates for the Betti number. Finally, for comparison, we have analyzed random subsets of the SAW, showing them to have the same fractal dimension as the SAW.