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Reviews of topical problems


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


Landau Institute for Theoretical Physics, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 119334, Russian Federation

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.

Fulltext pdf (1.1 MB)
Fulltext is also available at DOI: 10.1070/PU1998v041n04ABEH000382
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)
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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

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