STATISTICAL TOPOLOGY OF RANDOM WALKS

Serguei NECHAEV

Laboratoire de physique théorique et de modèles statistiques, Universite Paris-Sud

We discuss few interlinked topics in statistics of entangled random walks: conformal methods in topology of random path on multi-punctured plane, random walks on graphs and groups (including braid groups), "matrix-valued" Brownian bridges and random walks in Lobachevsky geometry. We explain how all these subjects help in understanding topology and fractal structure of strongly collapsed unknotted ring polymer chain.
[1] S. Nechaev. Statistics of knots and entangled random walks, Lectures presented at Les Houches. In: Summer School "Topological Aspects of Low Dimensional Systems", July 7--31, 1998. NATO Advanced Study Institute, session LXIX: EDP Sciences; Springer, 1999.
[2] S. Nechaev, O. Vasilyev. Thermodynamics and topology of disordered knots: correlations in trivial lattice knot diagrams. In: Physical and Numerical Models in Knot Theory. Series on Knots and Everything. WSPC: Singapore, 2005, chapter 22, pp. 421-472.
[3] M. Imakaev, L. Mirny, S. Nechaev. Effects of topological constraints on globular polymers. Soft Matter, 11 (2015) 665-671.
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