Advertisement

Probability Theory and Related Fields

, Volume 173, Issue 3–4, pp 1265–1299 | Cite as

On pinned fields, interlacements, and random walk on \(({\mathbb {Z}}/N {\mathbb {Z}})^2\)

  • Pierre-François RodriguezEmail author
Article
  • 73 Downloads

Abstract

We define two families of Poissonian soups of bidirectional trajectories on \({\mathbb {Z}}^2\), which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus \(({\mathbb {Z}}/N {\mathbb {Z}})^2\), started from the uniform distribution, run up to a time of order \((N\log N)^2\) and forced to avoid a fixed point. The local limit of the latter was recently established in Comets et al. (Commun Math Phys 343:129–164, 2016). Our construction proceeds by considering, somewhat in the spirit of statistical mechanics, a sequence of “finite volume” approximations, consisting of random walks avoiding the origin and killed at spatial scale N, either using Dirichlet boundary conditions, or by means of a suitably adjusted mass. By tuning the intensity u of such walks with N, the occupation field can be seen to have a nontrivial limit, corresponding to that of the actual random walk. Our construction thus yields a two-dimensional analogue of the random interlacements model introduced in Sznitman (Ann Math 171(3):2039–2087, 2010) in the transient case. It also links it to the pinned free field in \({\mathbb {Z}}^2\), by means of a (pinned) Ray–Knight type isomorphism theorem.

Mathematics Subject Classification

60F05 60G15 60K35 

References

  1. 1.
    Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math. 35(2), 209–273 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benjamini, I., Sznitman, A.-S.: Giant component and vacant set for random walk on a discrete torus. J. Eur. Math. Soc. 10(1), 133–172 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Černý, J., Teixeira, A.: From random walk trajectories to random interlacements. Ensaios Matemáticos, vol. 23. Sociedade Brasileira de Matemática, Rio de Janeiro (2012)Google Scholar
  4. 4.
    Černý, J., Teixeira, A.: Random walks on torus and random interlacements: macroscopic coupling and phase transition. Ann. Appl. Probab. 26(5), 2883–2914 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Comets, F., Popov, S.: The vacant set of two-dimensional critical random interlacement is infinite. Ann. Probab. 45(6B), 4752–4785 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Comets, F., Popov, S., Vachkovskaia, M.: Two-dimensional random interlacements and late points for random walks. Commun. Math. Phys. 343, 129–164 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dembo, A., Peres, Y., Rosen, J., Zeitouni, O.: Cover times for Brownian motion and random walks in two dimensions. Ann. Math. 160, 433–464 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Drewitz, A., Ráth, B., Sapozhnikov, A.: Local percolative properties of the vacant set of random interlacements with small intensity. Ann. Inst. Henri Poincaré Probab. Stat. 50(4), 1165–1197 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lawler, G.F.: Intersections of Random Walks. Probability and Its Applications. Birkhäuser, Boston (1991)CrossRefzbMATHGoogle Scholar
  10. 10.
    Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction, vol. 123. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Lupu, T.: From loop clusters and random interlacements to the free field. Ann. Probab. 44(3), 2117–2146 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Marcus, M.B., Rosen, J.: Markov Processes, Gaussian Processes, and Local Times, volume 100 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2006)Google Scholar
  13. 13.
    Resnick, S.I.: Extreme values, regular variation, and point processes. Applied Probability, vol. 4. A Series of the Applied Probability Trust. Springer, New York (1987)Google Scholar
  14. 14.
    Rodriguez, P.-F., Sznitman, A.-S.: Phase transition and level-set percolation for the Gaussian free field. Commun. Math. Phys. 320(2), 571–601 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Spitzer, F.: Principles of Random Walk. Graduate Texts in Mathematics, Vol. 34, 2nd edn. Springer, New York (1976)Google Scholar
  16. 16.
    Sznitman, A.-S.: Random walks on discrete cylinders and random interlacements. Probab. Theory Relat. Fields 145(1–2), 143–174 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sznitman, A.-S.: Vacant set of random interlacements and percolation. Ann. Math. 171(3), 2039–2087 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sznitman, A.-S.: Decoupling inequalities and interlacement percolation on \(G\times {\mathbb{Z}}\). Invent. Math. 187(3), 645–706 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sznitman, A.-S.: An isomorphism theorem for random interlacements. Electron. Commun. Probab. 17(9), 1–9 (2012)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Sznitman, A.-S.: On \(({\mathbb{Z}}/N{\mathbb{Z}})^2\)-occupation times, the Gaussian free field, and random interlacements. Bull. Inst. Math. Acad. Sin. (N.S.) 7(4), 565–602 (2012)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Sznitman, A.-S.: Random interlacements and the Gaussian free field. Ann. Probab. 40(6), 2400–2438 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sznitman, A.-S.: Topics in Occupation Times and Gaussian Free Fields. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zurich (2012)Google Scholar
  23. 23.
    Teixeira, A.: Interlacement percolation on transient weighted graphs. Electron. J. Probab. 14(54), 1604–1628 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Teixeira, A., Windisch, D.: On the fragmentation of a torus by random walk. Commun. Pure Appl. Math. 64(12), 1599–1646 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Windisch, D.: Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13, 140–150 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA

Personalised recommendations