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

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/
Web of Science

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

References (93) Cited by (8) Similar articles (20) ↓

V.S. Dotsenko “Universal randomness” Phys. Usp. 54 259–280 (2011)
L.M. Zelenyi, A.V. Milovanov “Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics” Phys. Usp. 47 749–788 (2004)
M.D. Frank-Kamenetskii, A.V. Vologodskii “Topological aspects of the physics of polymers: The theory and its biophysical applications” Sov. Phys. Usp. 24 679–696 (1981)
Ya.A. Smorodinskii, A.L. Shelepin, L.A. Shelepin “Groups and probabilities at the foundations of quantum mechanics” Sov. Phys. Usp. 35 (12) 1005–1051 (1992)
M.V. Vol’kenshtein “Physics and biology” Sov. Phys. Usp. 16 207–216 (1973)
B.K. Vainshtein “Three-dimensional electron microscopy of biological macromolecules” Sov. Phys. Usp. 16 185–206 (1973)
G.M. Zaslavskii, B.V. Chirikov “Stochastic instability of non-linear oscillations” Sov. Phys. Usp. 14 549–568 (1972)
A.R. Khokhlov, E.E. Dormidontova “Self-organization in ion-containing polymer systems” Phys. Usp. 40 109–124 (1997)
A.G. Kyurkchan, B.Yu. Sternin, V.E. Shatalov “Singularities of continuation of wave fields” Phys. Usp. 39 1221–1242 (1996)
V.I. Alkhimov “Excluded volume effect in statistics of self-avoiding walks” Phys. Usp. 37 527–561 (1994)
V.I. Alkhimov “The self-avoiding walk problem” Sov. Phys. Usp. 34 (9) 804–816 (1991)
A.N. Semenov, A.R. Khokhlov “Statistical physics of liquid-crystalline polymers” Sov. Phys. Usp. 31 988–1014 (1988)
M.A. Ol’shanetskii “A short guide to modern geometry for physicists” Sov. Phys. Usp. 25 123–129 (1982)
A.D. Dolgov “Massive and supermassive black holes in the contemporary and early Universe and problems in cosmology and astrophysics” Phys. Usp. 61 115–132 (2018)
G.N. Sarkisov “Approximate equations of the theory of liquids in the statistical thermodynamics of classical liquid systems” Phys. Usp. 42 545–561 (1999)
M.I. Tribel’skii “Short-wavelength instability and transition to chaos in distributed systems with additional symmetry” Phys. Usp. 40 159–180 (1997)
G.N. Chuev “Statistical physics of the solvated electron” Phys. Usp. 42 149 (1999)
V.I. Klyatskin, D. Gurarie “Coherent phenomena in stochastic dynamical systems” Phys. Usp. 42 165 (1999)
A.Yu. Grosberg “Disordered polymers” Phys. Usp. 40 125–158 (1997)
M. Birau, M.A. Krasil’nikov et al “Problems in the theory of relativistic plasma microwave electronics” Phys. Usp. 40 975–992 (1997)

The list is formed automatically.