Conformal Geometry and Invariants of 3-strand Brownian Braids

Sergei K. Nechaev 1, Raphael Voituriez 2

Nuclear Physics B 714 (2005) 336-356

We propose a simple geometrical construction of topological invariants of 3-strand Brownian braids viewed as world lines of 3 particles performing independent Brownian motions in the complex plane z. Our construction is based on the properties of conformal maps of doubly-punctured plane z to the universal covering surface. The special attention is paid to the case of indistinguishable particles. Our method of conformal maps allows us to investigate the statistical properties of the topological complexity of a bunch of 3-strand Brownian braids and to compute the expectation value of the irreducible braid length in the non-Abelian case.

  • 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 2. Laboratoire de Physique Théorique des Liquides (LPTL),
    CNRS : UMR7600 – Université Paris VI - Pierre et Marie Curie