Problems of probabilistic topology: the statistics of knots and non-commutative random walks

Abstract

The paper reviews the state of affairs in a 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 limiting behavior of random walks in multiconnected spaces and on noncommutative groups related to the knot theory. We discuss the application of the above- mentioned problems in the statistical physics of polymer chains. On the basis of noncommutative probability theory we derive some new results in the statistical physics of entangled polymer chains which unite rigorous mathematical facts with intuitive physical arguments