This 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 stimate the probability of trivial knot formation on a 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 non-commutative groups related to knot theory. We discuss the application of the above-mentioned problems in the statistical physics of polymer chains. On the basis of non-commutative probability theory we derive some new results in the statistical physics of entangled polymer chains which unite rigorous mathematical facts with intuitive physical arguments.

PACS: 02.40.Re, 02.50.Cw, 36.20.Ey (all) DOI: 10.1070/PU1998v041n04ABEH000382 URL: https://ufn.ru/en/articles/1998/4/a/ 000074653300001 Citation: Nechaev S K "Problems of probabilistic topology: the statistics of knots and non-commutative random walks" Phys. Usp. 41 313–347 (1998) BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks Оригинал: Нечаев С К «Проблемы вероятностной топологии: статистика узлов и некоммутативных случайных блужданий» УФН 168 369–405 (1998); DOI: 10.3367/UFNr.0168.199804a.0369