RSS feeds
РусскийEnglish
Issue 12, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

1998

 / 

April

←  →

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)
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)

  1. Frank-Kamenetskiî M D, Vologodskiî A V Usp. Fiz. Nauk134 641 (1981) [Sov. Phys. Usp. 24 679 (1981)]; Vologodskiî A V et al. Zh. Exp. Teor. Fiz.66 2153 (1974) [Sov. Phys. JETP 39 1059 (1974)]; Vologodskiî A V, Lukashin A V, Frank-Kamenetskiî M D Zh. Exp. Teor. Fiz.67 1875 (1974) [Sov. Phys. JETP 40 932 (1975)]; Frank-Kamenetskii M D, Lukashin A V, Vologodskii A V Nature (London) 258 398 (1975)
  2. Wilsceck F Fractional Statistics and Anyon Superconductivity (Singapore: WSPC, 1990)
  3. Majid S Int. J. Mod. Phys. A 8 4521 (1993)
  4. Gómez C, Sierra G J. Math. Phys.34 2119 (1993)
  5. Jones V F R Pacific J. Math.137 311 (1989); Likorish W B R Bull. London Math. Soc. 20 558 (1988); Wadati M, Deguchi T, Akutsu Y Phys. Rep.180 247 (1989)
  6. Alekseev A et al. (Eds) Integrable Models and Strings (Lect. Not. Phys.; 436, Berlin, New York: Springer-Verlag, 1994); Kauffman L H AMS Contemp. Math. Series78 263 (1989)
  7. Kauffman L H Topology26 395 (1987)
  8. Vasil’ev V A Complements of Discriminants of Smooth Maps: Topology and Applications (Translations of Mathematical Monographs, Vol. 98, Providence, R.I.: American Mathematical Society, 1992); Bar-Natan D, Preprint Harvard, 1992
  9. Edwards S F Proc. Phys. Soc. London91 513 (1967)
  10. Yor M J. Appl. Prob.29 202 (1992); Math. Finance3 231 (1993)
  11. Comtet A, Desbois J, Monthus C J. Stat. Phys.73 433 (1993); Desbois J J. Phys. A25 L195, L755 (1992); Desbois J, Comtet A J. Phys. A25 3097 (1992)
  12. Wiegel F Introduction to Path-Integrals Methods in Physics and Polymer Science (Singapore: WSPC, 1986)
  13. Khandekar D C, Wiegel F W J. Phys. A21 L563 (1988); J. Phys. (Paris)50 263 (1989)
  14. Hofstadter D R Phys. Rev. B14 2239 (1976); Guillement J P, Helffer B, Treton P J. Phys. (Paris)50 2019 (1989)
  15. Wiegmann P B, Zabrodin A V Phys. Rev. Lett. 72 1890 (1994); Nucl. Phys. B422 495 (1994)
  16. Bellissard J, in Statistical Mechanics and Field Theory: Mathematical Aspects: Proc. of the International Conference (Groningen, Netherlands 26-30 August 1985, Lecture Notes in Physics, Vol. 257, Berlin: Springer-Verlag, 1986) p. 99
  17. Dubrovin B A, Novikov S P, Fomenko A T Sovremennaya Geometriya: Metody i Prilozheniya (Modern Geometry: Metods and Applications, Moscow: Nauka, 1979) [Translated into English (New York: Springer-Verlag, 1984)]
  18. Wu F Y Rev. Mod. Phys.54 235 (1982)
  19. Baxter R J Exactly Solved Models in Statistical Mechanics (London, New York: Academic Press, 1982)
  20. Mezard M, Parisi G, Virasoro M Spin Glass Theory and Beyond (Singapore: WSPC, 1987)
  21. Lifshits I M, Gredeskul S A, Pastur L A Vvedenie v Theoriyu Neuporyadochennykh Sistem (Introduction to the theory of disordered systems, Moscow: Nauka, 1982) [Translated into English (New York: Wiley, 1988)]
  22. Reidemeister K Knotentheorie (Berlin: Springer, 1932)
  23. Kauffman L H Knots and Physics (Teaneck, NJ: World Scientific, 1991)
  24. Grosberg A Yu, Nechaev S J. Phys. A25 4659 (1992)
  25. Wu F Y J. Knot Theory Ramific.1 32 (1992)
  26. Grosberg A Yu, Nechaev S Europhys. Lett.20 613 (1992)
  27. Duplantier B, David F J. Stat. Phys.51 327 (1988)
  28. Edwards S F, Anderson P W J. Phys. F5 965 (1975)
  29. Cwilich G, Kirkpatrick T R J. Phys. A22 4971 (1989)
  30. De Santis E, Parisi G, Ritort F, Preprint: cond-mat/9410093
  31. Gross D J, Kanter I, Sompolinsky H Phys. Rev. Lett.55 304 (1985)
  32. Bellissard J, Simon B J. Funct. Anal.48 408 (1982)
  33. Gutzwiller M Chaos in Classical and Quantum Mechanics (New York: Springer-Verlag, 1990)
  34. Terras A Harmonic Analysis on Symmetric Spaces and Applications (New York: Springer-Verlag, 1985-1988)
  35. Hejhal D A The Selberg Trace Formula for PSL(2,IR, Lect. Notes in Math., 548, Berlin, New York: Springer-Verlag, 1976); (Lect Notes in Math, 1001, Berlin, New York: Springer-Verlag, 1983)
  36. Chassaing P, Letac G, Mora M Probability Measures on Groups (Lect. Notes Math., 1064, Berlin, New York: Springer-Verlag, 1983)
  37. Nechaev S K Statistics of Knots and Entangled Random Walks (Piver Edge, NJ: World Scientific, 1996)
  38. Lerda A Anyons: Quantum Mechanics of Particles with Fractional Statistic (Lect. Notes in Phys. New Series m, m 14, Berlin, New York: Springer-Verlag, 1992)
  39. Fürstenberg H Trans. Am. Math. Soc.198 377 (1963); Tutubalin V Teor. Ver. Primen. 10 19 (1965) [Theory Prob. Appl. 10 32 (1965)]
  40. Nechaev S, Sinai Ya Bol. Sci. Bras. Mat.21 121 (1991)
  41. Koralov L, Nechaev S, Sinai Ya Prob. Theor. Appl.38 331 (1993)
  42. Mirlin A, Fyodorov Y J. Phys. A24 227 (1991); Fyodorov Y Basic Features of Efetov’s supersymmetry approach Mesoscopic Quantum Physics, Les Houches, session LXI, 1994 (Amstredam: Elsevier, 1995)
  43. Khokhlov A R, Nechaev S K Phys. Lett. A112 156 (1985); Nechaev S K, Semenov A N, Koleva M K Physica A140 506 (1987)
  44. Nechaev S, Vershik A J. Phys. A27 2289 (1994)
  45. Monastyrsky M, Nechaev S Mod. Phys. Lett. A12 589 (1997)
  46. Nechaev S K, Grosberg A Yu, Vershik A M J. Phys. A29 2411 (1996)
  47. Kesten H Trans. Amer. Math. Soc.92 336 (1959)
  48. Vershik A M Topics in Algebra26 pt. 2 467 (1990); Proc. Am. Math. Soc.148 1 (1991)
  49. Helfand E, Pearson D S J. Chem. Phys.79 2054 (1983); Rubinstein M, Helfand E J. Chem. Phys.82 2477 (1985)
  50. Cates M E, Deutsch J M J. Phys. (Paris)47 2121 (1986)
  51. Vershik A M private communication
  52. Desbois J, Nechaev S J. Stat. Phys.88 201 (1997)
  53. Comtet A, Nechaev S Nucl. Phys. B in press, preprint cond-mat/9707120
  54. Desbois J, Nechaev S J. Phys. A in press, preprint cond-mat/9707121
  55. Jones V F R Bull. Am. Math. Soc.12 103 (1985)
  56. Vershik A M, Kerov S V Dokl. Acad. Nauk SSSR301 777 (1988)
  57. Kauffman L H, Saleur H Comm. Math. Phys. 141 293 (1991)
  58. Akutsu Y, Deguchi T Phys. Rev. Lett.67 777 (1991); Akutsu Y, Wadati M J. Phys. Soc. Jpn.56 839, 3039 (1987)
  59. Birman J Knots, Links and Mapping Class Groups (Ann. Math. Stud., 82, Princeton, NJ: Princeton Univ. Press, 1976)
  60. Mumford M Tata Lectures on Theta, I, II (Boston, MA: Birkhauser, 1983)
  61. Letchikov A Usp. Math. Nauk 51 (1) 51 (1996) [Russian Math. Surveys 51 49 (1996)]
  62. Zinn-Justin J Quantum Field Theory and Critical Phenomena (Oxford: Clarendon Press; New York: Oxford University Press, 1989)
  63. Gerzenshtein M E, Vasiljev V B Teor. Ver. Prim. 4 424 (1959) [Theor. Prob. Appl. 4 432 (1959)]; Karpelevich F I, Tutubalin V N, Shour M G Teor. Ver. Prim. 4 432 (1959) [Theor. Prob. Appl.4 436 (1959)]
  64. Polyakov A M Mod. Phys. Lett. A3 325 (1988); Witten E Comm. Math. Phys.121 351 (1989)
  65. Belavin A A, Polyakov A M, Zamolodchikov A B Nucl. Phys. B241 333 (1984)
  66. Nechaev S K J. Phys. A21 3659 (1988)
  67. Ito K, McKean H P Diffusion Processes and Their Sample Paths (Berlin, New York: Springer-Verlag, 1965)
  68. Golubev V V Lektsii po Analiticheskoî Teorii Differentsial’nykh Uravneniî (Lectures on Analytic Theory of Differential Equations, Moscow: GITTL, 1950); Venkov A B, Preprint LOMI P/2/86
  69. Dotsenko Vl S Nucl. Phys. B235 [FS11] 54 (1984)
  70. Knizhnik V G, Zamolodchikov A B Nucl. Phys. B 247 83 (1984)
  71. Guadarini E, Martinelli M, Mintchev M, CERN-TH preprints 5419/89, 5420/89, 5479/89
  72. Chandrassekharan K Elliptic Functions (Berlin: Springer, 1985)
  73. Grosberg A Yu, Khokhlov A R Statisticheskaya Fizika Makromolekul (Statistical Physics of Macromolecules, Moscow: Nauka, 1993) [Translated into English (New York: AIP Press, 1994)]
  74. Koralov L B, Nechaev S K, Sinai Ya G Chaos1 131 (1991)
  75. Gutin A M, Grosberg A Yu, Shakhnovich E I Macromolecules26 1293 (1993)
  76. De Gennes P G Scaling Concepts in Polymer Physics (Ithaca, N.Y.: Cornell Univ. Press, 1979)
  77. Kholodenko A, private communications
  78. Lifshitz I M, Grosberg A Yu, Khokhlov A R Rev. Mod. Phys.50 683 (1978)
  79. Grosberg A Yu, Nechaev S K, Shakhnovich E I J. Phys. (Paris)49 2095 (1988)
  80. Chu B, Ying Q, Grosberg A Macromolecules28 180 (1995)
  81. Grosberg A, Rabin Y, Havlin S, Neer A Europhys. Lett.23 373 (1993); Grosberg A, Rabin Y, Havlin S, Neer A Biofizika38 75 (1993)
  82. Ma J, Straub J E, Shakhnovich E I J. Chem. Phys. 103 2615 (1995)
  83. Grosberg A Yu, Nechaev S K Macromolecules24 2789 (1991)
  84. Khokhlov A R, Nechaev S K J. Phys. II (Paris) 6 1547 (1996)
  85. Nachlis W L, Kambour R P, McKnight W J Abstracts of the 35th IUPAC Congress2 796; Polymer39 3643 (1994)
  86. Michels J P J, Wiegel F W Proc. R. Soc. London Ser. A403 269 (1986)
  87. Koniaris K, Muthukumar M Phys. Rev. Lett.66 2211 (1991)
  88. Tsurusaki K, Deguchi T J. Phys. Soc. Jpn.64 1506 (1995); Deguchi T, Tsurusaki K J. Knot Theory Ramific.3 321 (1994)
  89. Klenin K V, Vologodskî A V, Anshelevich V V, Dykhne A M J. Biomol. Struct. Dyn.5 1173 (1988)
  90. Van Rensburg E J, Whittington S G J. Phys. A 23 3573 (1990); Le Bret M Biopolymers19 619 (1980)
  91. Fixman M J. Chem. Phys.36 306 (1962)
  92. Lifshits I M, Grosberg A Yu Zh. Exp. Teor. Fiz.65 2399 (1973) [Sov. Phys. JETP 38 1198 (1974)]
  93. Grosberg A Yu, Nechaev S K Adv. Polym. Sci. 106 1 (1993)
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions