Skip to main content
SpringerLink
Log in

Explicit Formulae in Probability and in Statistical Physics

  • Chapter
Book cover In Memoriam Marc Yor - Séminaire de Probabilités XLVII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2137))

Abstract

We consider two aspects of Marc Yor’s work that have had an impact in statistical physics: firstly, his results on the windings of planar Brownian motion and their implications for the study of polymers; secondly, his theory of exponential functionals of Lévy processes and its connections with disordered systems. Particular emphasis is placed on techniques leading to explicit calculations.

2010Mathematics Subject Classification: 60-02, 82B44

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. P.W. Anderson, Absence of diffusion in a certain random lattice. Phys. Rev. 109, 1492–1505 (1958)

    Article  Google Scholar 

  2. F. Baudoin, N. O’Connell, Exponential functionals of Brownian motion and class-one Whittaker functions. Ann. Inst. Henri Poincaré Probab. Stat. 47, 1096–1120 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Bélisle, Windings of random walks. Ann. Probab. 17, 1377–1402 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  4. M.A. Berger, P.H. Roberts, On the number winding problem with finite steps. Adv. Appl. Probab. 20, 261–274 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Bertoin, Lévy Processes (Cambridge University Press, Cambridge, 1996)

    MATH  Google Scholar 

  6. J. Bertoin, D. Dufresne, M. Yor, Some two-dimensional extensions of Bougerol’s identity in law for the exponential functional of linear Brownian motion. Rev. Mat. Iberoam. 29, 1307–1324 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Bertoin, M. Yor, Exponential functionals of Lévy processes. Probab. Surv. 2, 191–212 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. J.-P. Bouchaud, A. Comtet, A. Georges, P. Le Doussal, Classical diffusion of a particle in a one-dimensional random force field. Ann. Phys. 201, 285–341 (1990)

    Article  Google Scholar 

  9. C. Bouchiat, M. Mézard, Elasticity model of a supercoiled DNA molecule. Phys. Rev. Lett. 80, 1556–1559 (1998)

    Article  Google Scholar 

  10. P. Bougerol, Examples de théorèmes locaux sur les groupes résolubles. Ann. Inst. H. Poincaré Sect. B (N.S.) 19, 403–432 (1983)

    Google Scholar 

  11. P. Bougerol, J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators (Birkhaüser, Basel, 1985)

    Book  MATH  Google Scholar 

  12. P. Carmona, F. Petit, M. Yor, On the distribution and asymptotic results for exponential functionals of Lévy processes, in Exponential functionals and principal values related to Brownian motion. Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, pp. 73–130 (1997)

    Google Scholar 

  13. A. Comtet, J. Desbois, C. Monthus, Asymptotic laws for the winding angles of planar Brownian motion. J. Stat. Phys. 73, 433–440 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Comtet, J. Desbois, S. Ouvry, Winding of planar Brownian curves. J. Phys. A Math. Gen. 23, 3563–3572 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Comtet, C. Texier, Y. Tourigny, Supersymmetric quantum mechanics with Lévy disorder in one dimension. J. Stat. Phys. 145, 1291–1323 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Comtet, C. Texier, Y. Tourigny, Lyapunov exponents, one-dimensional Anderson localization and products of random matrices. J. Phys. A Math. Theor. 46, 254003 (2013)

    Article  MathSciNet  Google Scholar 

  17. A. Comtet, C. Texier, Y. Tourigny, Products of random matrices and generalised quantum point scatterers. J. Stat. Phys. 140, 427–466 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. J. Desbois, C. Furtlehner, S. Ouvry, Random magnetic impurities and the Landau problem. Nucl. Phys. B 453 [FS], 759–776 (1995)

    Google Scholar 

  19. B. Drossel, M. Kardar, Winding angle distributions for random walks and flux lines. Phys. Rev. E 53, 5861–5871 (1996)

    Article  Google Scholar 

  20. D. Dufresne, The distribution of a perpetuity, with applications to risk theory and pension funding. Scan. Actuar. J. no. 1–2, 39–79 (1990)

    Article  MathSciNet  Google Scholar 

  21. F.J. Dyson, The dynamics of a random linear chain. Phys. Rev. 92, 1331–1338 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  22. S.F. Edwards, Statistical mechanics with topological constraints I. Proc. Phys. Soc 91, 513–519 (1967)

    Article  MATH  Google Scholar 

  23. H.L. Frisch, S.P. Lloyd, Electron levels in a one-dimensional lattice. Phys. Rev. 120, 1175–1189 (1960)

    Article  MATH  Google Scholar 

  24. C. Garban, J.A. Trujillo-Ferreras, The expected area of the filled planar Brownian loop is π∕5. Commun. Math. Phys. 264, 797–810 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. G. Gavazzi, J.M. Wheatley, A.J. Schofield, Single particle motion in a random magnetic flux. Phys. Rev. E 47, 15170–15176 (1993)

    Article  Google Scholar 

  26. H.K. Gjessing, J. Paulsen, Present value distribution with applications to ruin theory and stochastic equations. Stoch. Process. Appl. 71, 123–144 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  27. A. Grosberg, H. Frisch, Winding angle distribution for planar random walk, polymer ring entangled with an obstacle, and all that: Spitzer–Edwards–Prager–Frisch model revisited. J. Phys. A Math. Gen. 36, 8955 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. F. Hirsch, M. Yor, On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator. Bernoulli 19, 1350–1377 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. G. Kallianpur, H. Robbins, Ergodic property of the Brownian motion process. Proc. Nat. Acad. Sci. USA 39, 525–533 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  30. J.F.C. Kingman, Subadditive ergodic theory. Ann. Probab. 1, 883–899 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  31. S. Kotani, On asymptotic behaviour of the spectra of a one-dimensional Hamiltonian with a certain random coefficient. Publ. RIMS Kyoto Univ. 12, 447–492 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  32. J. Lamperti, Semi–stable Markov processes I. Z. Wahrsch. verw. Geb. 22, 205–225 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  33. J.-M. Luck, Systèmes désordonnés unidimensionels (Aléa, Saclay, 1992)

    Google Scholar 

  34. H. Matsumoto, M. Yor, Exponential functionals of Brownian motion I: probability laws at fixed times. Probab. Surv. 2, 312–347 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  35. H. Matsumoto, M. Yor, Exponential functionals of Brownian motion II: some related diffusion processes. Probab. Surv. 2, 348–384 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  36. P. Messulam, M. Yor, On D. Williams’ “pinching method” and some applications. J. Lond. Math. Soc. s2–26, 348–364 (1982)

    Google Scholar 

  37. D.R. Nelson, Vortex entanglement in high-T c superconductors. Phys. Rev. Lett. 60, 1973–1976 (1988)

    Article  MathSciNet  Google Scholar 

  38. N. O’Connell, Directed polymers and the quantum Toda lattice. Ann. Probab. 40, 437–358 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  39. B. Øksendal, Stochastic Differential Equations, 6th edn. (Springer, Berlin, 2010)

    Google Scholar 

  40. L.A. Pastur, A. Figotin, Spectra of Random and Quasi-periodic Operators (Springer, Berlin, 1992)

    Book  Google Scholar 

  41. P. Patie, Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes. Ann. Inst. H. Poincaré Probab. Stat. 45, 667–684 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  42. J. Pitman, M. Yor, Asymptotic laws of planar Brownian motion. Ann. Probab. 14, 733–779 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  43. S. Prager, H.L. Frisch, Statistical mechanics of a simple entanglement. J. Chem. Phys. 46, 1475–1483 (1967)

    Article  Google Scholar 

  44. D. Revuz, M. Yor, Continuous Martingales and Brownian Motion (Springer, New-York, 1999)

    Book  MATH  Google Scholar 

  45. J. Rudnick, Y. Hu, The winding angle distribution of an ordinary random walk. J. Phys. A Math. Gen. 20, 4421–4439 (1987)

    Article  MathSciNet  Google Scholar 

  46. H. Saleur, Winding angle distribution for Brownian and self-avoiding walks. Phys. Rev. E 50, 1123–1130 (1994)

    Article  Google Scholar 

  47. F. Spitzer, Some theorems concerning two-dimensional Brownian motions. Trans. Am. Math. Soc. 87, 187–197 (1958)

    MATH  MathSciNet  Google Scholar 

  48. S. Vakeroudis, M. Yor, D. Holcman, The mean first rotation time of a planar polymer. J. Stat. Phys. 143, 1074–1095 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  49. W. Werner, Formule de Green, lacet brownien plan et aire de Lévy. Stoch. Process. Appl. 57, 225–245 (1995)

    Article  MATH  Google Scholar 

  50. M. Yor, Loi de l’indice du lacet Brownien et distributions de Hartman–Watson. Z. Wahrsch. Verw. Gebiete 53, 71–95 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  51. M. Yor, Exponential Functionals of Brownian Motion and Related Processes (Springer, Berlin, 2001)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

It is a pleasure to thank our colleague Christophe Texier for commenting on the manuscript.

Author information

Authors and Affiliations

  1. UPMC Univ. Paris 6, 75005, Paris, France

    Alain Comtet

  2. Univ. Paris Sud, CNRS, LPTMS, UMR 8626, Orsay, 91405, France

    Alain Comtet

  3. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Yves Tourigny

Authors
  1. Alain Comtet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yves Tourigny
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alain Comtet .

Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles, France

    Catherine Donati-Martin

  2. Campus scientifique, IECL, Vandœuvre-lès-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St Quentin, Versailles, France

    Alain Rouault

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Comtet, A., Tourigny, Y. (2015). Explicit Formulae in Probability and in Statistical Physics. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) In Memoriam Marc Yor - Séminaire de Probabilités XLVII. Lecture Notes in Mathematics(), vol 2137. Springer, Cham. https://doi.org/10.1007/978-3-319-18585-9_22

Download citation

Publish with us

Policies and ethics

Search

Navigation