Statistics of knots and entangled random walks

TL;DR

This paper explores the probability of knot formation in random walks and polymers, linking probabilistic topology with statistical physics, and introduces new mathematical insights into entangled polymer behavior.

Contribution

It connects knot theory with statistical physics of polymers, providing new results on entanglement using non-commutative probability and lattice invariants.

Findings

Probability estimates for trivial knots using Kauffman invariants

Connection between knot formation and 2D Potts model thermodynamics

New results on polymer entanglement from non-commutative probability

Abstract

The lectures review the state of affairs in modern branch of mathematical physics called probabilistic topology. In particular we consider the following problems: (i) We estimate the probability of a trivial knot formation on the lattice using the Kauffman algebraic invariants and show the connection of this problem with the thermodynamic properties of 2D disordered Potts model; (ii) We investigate the limit behavior of random walks in multi-connected spaces and on non-commutative groups related to the knot theory. We discuss the application of the above mentioned problems in statistical physics of polymer chains. On the basis of non-commutative probability theory we derive some new results in statistical physics of entangled polymer chains which unite rigorous mathematical facts with more intuitive physical arguments.