Topological localization in unentangled random braids

par Sergei Nechaev (LPTMS - CNRS & Universite Paris Saclay)

vendredi 13 nov. 2026, 15:00 → 17:00 Europe/Paris

Salle Marie-Louise Dubreil-Jacotin (IHP - Bâtiment Borel)

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 tread in a braid are qualitatively different from those of a thread embedded in a medium of fixed parallel lines. In particular, they observed a logarithmic increase in the transverse span, D, of a thread with its length, N. Studying this problem both numerically and analytically, we propose the following interpretation: the logarithmic regime is an intermediate one, and sufficiently long threads exhibit topological localization, where span D approaches a finite limit at N >> 1. Unlike localization induced by quenched external disorder, this phenomenon arises solely from collective topological constraints, giving rise to topological microphase separation. We study braids within a locally free approximation and confirm the localization by analytical investigation of Brownian bridges on the asymptotic Cayley graph of a locally free group LF_n which approximates the braid group B_{n+1}. We conjecture that topologically induced localization emerges as a phase transition: for a small number of threads, n ~ 3 -- 5, the span, D, grows algebraically with N (N>>1), as D ~ N^{1/4}. By contrast, for large n, each thread in a long braid asymptotically localizes, approaching D ~ const.