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 1998 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", chapter 22, pp. 421-472, Series on Knots and Everything, (WSPC: Singapore, 2005)
[3] M. Imakaev, L. Mirny, S. Nechaev, Effects of topological constraints on globular polymers, Soft Matter 11 (2015), 665-671
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