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Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 775–818 | Cite as

The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph

  • Titus Lupu
  • Wendelin Werner
Article

Abstract

We further investigate properties of the Gaussian free field (GFF) on the metric graph associated to a discrete weighted graph (where the edges of the latter are replaced by continuous line-segments of appropriate length) that has been introduced by the first author. On such a metric graph, the GFF is a random continuous function that generalises one-dimensional Brownian bridges so that one-dimensional techniques can be used. In the present paper, we define and study the pseudo-metric defined on the metric graph (and therefore also on the discrete graph itself), where the length of a path on the metric graph is defined to be the local time at level zero accumulated by the Gaussian free field along this path. We first derive a pathwise transformation that relates the GFF on the metric graph with the reflected GFF on the metric graph via the pseudo-distance defined by the latter. This is a generalisation of Paul Lévy’s result relating the local time at zero of Brownian motion to the supremum of another Brownian motion. We also compute explicitly the distribution of certain functionals of this pseudo-metric and of the GFF. In particular, we point out that when the boundary consists of just two points, the law of the pseudo-distance between them depends solely on the resistance of the network between them. We then discuss questions related to the scaling limit of this pseudo-metric in the two-dimensional case, which should be the conformally invariant way to measure distances between CLE(4) loops introduced and studied by the second author with Wu, and by Sheffield, Watson and Wu. Our explicit laws on metric graphs also lead to new conjectures for related functionals of the continuum GFF on fairly general Riemann surfaces.

Keywords

Random distances Gaussian free field Metric graph Local time Conformal loop ensemble 

Mathematics Subject Classification

60G60 60G15 60K35 60J55 60J65 60J67 

Notes

Acknowledgements

T.L. acknowledges the support of Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation. W.W. acknowledges the support of the SNF Grant SNF-155922. The authors are also part of the NCCR Swissmap of the SNF. They also thank the referees for their comments.

References

  1. 1.
    Ahlfors, L.V.: Conformal Invariants. Topics in Geometric Function Theory. AMS Chelsea Publishing, New York (2010)MATHGoogle Scholar
  2. 2.
    Aru, J., Sepulveda, A., Werner, W.: On bounded-type thin local sets of the two-dimensional Gaussian free fields. J. Inst. Math. Jussieu (2016). arXiv:1603.03362 (to appear)
  3. 3.
    Bertoin, J., Pitman, J.: Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118, 147–166 (1994)MathSciNetMATHGoogle Scholar
  4. 4.
    Borodin, A., Salminen, P.: Handbook of Brownian Motion: Facts and Formulae, Volume XIV of Probability and Its Applications. Birkhäuser, Basel (1996)CrossRefMATHGoogle Scholar
  5. 5.
    Ding J., Li, L.: Chemical distances for level-set percolation of two-dimensional discrete Gaussian free fields (2016). arXiv:1605.04449
  6. 6.
    Dubédat, J.: SLE and Virasoro representations: localization. Commun. Math. Phys. 336(2), 695–760 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Duffin, R.J.: The extremal length of a network. J. Math. Anal. Appl. 5, 200–215 (1962)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Duplantier, B., Sheffield, S.: Liouville quantum gravity and KPZ. Invent. Math. 185, 333–393 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Farkas, H.M., Kra, I.: Riemann Surfaces, Volume 71 of Graduate Text in Mathematics, 2nd edn. Springer, Berlin (1992)Google Scholar
  10. 10.
    Garaberdian, P.R.: Partial Differential Equations. Chelsia Publishing Company, New York (1986)Google Scholar
  11. 11.
    Jurchescu, M.: Bordered Riemann surfaces. Math. Ann. 143, 264–292 (1961)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, Volume 113 of Grad. Texts in Math, 2nd edn. Springer, Berlin (2010)Google Scholar
  13. 13.
    Kemppainen, A., Werner, W.: The nested simple conformal loop ensembles in the Riemann sphere. Probab. Theory Relat. Fields 165, 835–866 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lawler, G.F.: Partition functions, loop measure, and versions of SLE. J. Stat. Phys. 134, 813–837 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge Stud. Adv. Math, vol. 123. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
  16. 16.
    Lawler, G.F., Werner, W.: Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. 2, 291–328 (2000)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lawler, G.F., Werner, W.: The Brownian loop-soup. Probab. Theory Relat. Fields 128, 565–588 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Le Jan, Y.: Markov paths, loops and fields. In: 2008 St-Flour Summer School, L.N. Math., Vol. 2026. Springer, Berlin (2011)Google Scholar
  19. 19.
    Le Jan, Y., Marcus, M.B., Rosen, J.: Permanental fields, loop soups and continuous additive functionals. Ann. Probab. 43(1), 44–84 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lupu, T.: From loop clusters and random interlacements to the free field. Ann. Probab. 44(3), 2117–2146 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lupu,T.: Convergence of the two-dimensional random walk loop soup clusters to CLE. J. Eur. Math. Soc. (2015). arXiv:1502.06827 (to appear)
  22. 22.
    Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge University Press, Cambridge (2017)MATHGoogle Scholar
  23. 23.
    Miller, J., Sheffield, S.: Imaginary geometry I: interacting SLEs. Probab. Theory Relat. Fields 164, 553–705 (2016)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Miller, J., Sheffield, S.: Quantum Loewner evolution. Duke Math. J. 165(17), 3241–3378 (2016)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Miller, J., Sheffield, S.: CLE(4) and the Gaussian free field. In preparationGoogle Scholar
  26. 26.
    Miller, J., Watson, S.S., Wilson, D.B.: The conformal loop ensemble nesting field. Probab. Theory Relat. Fields 163, 769–801 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Qian, W., Werner, W.: Decomposition of Brownian loop-soup clusters (2015). arXiv:1509.01180
  28. 28.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Volume 293 of Grundlehren Math. Wiss, 3rd edn. Springer, Berlin (1999)CrossRefGoogle Scholar
  29. 29.
    Rozanov, Y.A.: Markov Random Fields, 1st edn. Springer, Berlin (1982)CrossRefMATHGoogle Scholar
  30. 30.
    Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202, 21–137 (2009)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Schramm, O., Sheffield, S.: A contour line of the continuum Gaussian free field. Probab. Theory Relat. Fields 157, 47–80 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Sheffield, S.: The Gaussian free field for mathematicians. Probab. Theory Relat. Fields 139, 521–541 (2007)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Sheffield, S., Watson, S.S., Wu, H.: A conformally invariant metric on CLE(4). In preparationGoogle Scholar
  34. 34.
    Sheffield, S., Werner, W.: Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Ann. Math. 176(3), 1827–1917 (2012)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Sznitman, A.-S.: Disconnection and level-set percolation for the Gaussian free field. J. Math. Soc. Japan 67(4), 1801–1843 (2015)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Wang, M., Wu, H.: Level lines of Gaussian free field I: zero-boundary GFF. Stoch. Process. Appl. 127(4), 1045–1124 (2017)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Werner, W.: Topics on the two-dimensional Gaussian free field. Lecture notes (2014)Google Scholar
  38. 38.
    Werner, W., Wu, H.: On conformally invariant CLE explorations. Commun. Math. Phys. 320, 637–661 (2013)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Wu, H.: Autour des relations entre SLE, CLE, champ libre gaussien, et conséquences. Ph.D. thesis, Ecole doctorale Mathématiques de la région Paris-Sud (2013)Google Scholar
  40. 40.
    Yor, M.: Some Aspects of Brownian Motion, Part I: Some Special Functionals. Birkhäuser, Basel (1992)MATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institute for Theoretical StudiesETH ZürichZurichSwitzerland
  2. 2.Department of MathematicsETH ZürichZurichSwitzerland

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