Topological localization in unentangled random braids
Abstract
In 2015, P. Serna, G. Bunin, and A. Nahum numerically investigated fluctuations in a bunch of unentangled random threads. They showed that fluctuations of an individual thread in a braid are qualitatively different from those of a thread embedded in a medium of fixed parallel lines. In particular, Serna et al. observed a logarithmic increase in the average transverse span, <|xN|>, of a thread with polymer length, N. This behavior highlights the collective nature of topological interactions, in which each polymer fluctuates and constrains the motion of the others. Studying this problem both numerically and analytically, we propose the following interpretation: the logarithmic regime is transient, and sufficiently long polymers exhibit topological localization, where <|xN|> approaches a finite limit at N→∞. Unlike localization induced by quenched external disorder, this phenomenon arises solely from collective topological constraints. We study braids within a locally free approximation and support the localization by analytical investigation of Brownian bridges on the asymptotic Cayley graph of a locally free group. We further conjecture that topologically induced localization emerges as a phase transition: for a small number of threads, n ∼ 2÷5, the span, <|xN|>, grows algebraically, with the usual scaling <|xN|>∼ N1/4 at N≫ 1; by contrast, for sufficiently large n, each thread in a long braid (N≫ 1) asymptotically localizes, approaching <|xN|>∼ const.
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BibTeX
@misc{fedotov2026topological,
title = {{Topological localization in unentangled random braids}},
author = {Daniil Fedotov and Pieter Hendrik Willem Van Der Hoek and Sergei Nechaev},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05721},
eprint = {2611.05721},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05721},
}
RIS
TY - UNPB
TI - Topological localization in unentangled random braids
AU - Daniil Fedotov
AU - Pieter Hendrik Willem Van Der Hoek
AU - Sergei Nechaev
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05721
UR - https://arxiv.org/abs/2611.05721
AB - In 2015, P. Serna, G. Bunin, and A. Nahum numerically investigated fluctuations in a bunch of unentangled random threads. They showed that fluctuations of an individual thread in a braid are qualitatively different from those of a thread embedded in a medium of fixed parallel lines. In particular, Serna et al. observed a logarithmic increase in the average transverse span, <|x_N|>, of a thread with polymer length, N. This behavior highlights the collective nature of topological interactions, in which each polymer fluctuates and constrains the motion of the others. Studying this problem both numerically and analytically, we propose the following interpretation: the logarithmic regime is transient, and sufficiently long polymers exhibit topological localization, where <|x_N|> approaches a finite limit at N→∞. Unlike localization induced by quenched external disorder, this phenomenon arises solely from collective topological constraints. We study braids within a locally free approximation and support the localization by analytical investigation of Brownian bridges on the asymptotic Cayley graph of a locally free group. We further conjecture that topologically induced localization emerges as a phase transition: for a small number of threads, n ∼ 2÷5, the span, <|x_N|>, grows algebraically, with the usual scaling <|x_N|>∼ N^1/4 at N≫ 1; by contrast, for sufficiently large n, each thread in a long braid (N≫ 1) asymptotically localizes, approaching <|x_N|>∼ const.
ER -