The Annals of Probability

From loop clusters and random interlacements to the free field

Titus Lupu

Full-text: Open access

Abstract

It was shown by Le Jan that the occupation field of a Poisson ensemble of Markov loops (“loop soup”) of parameter $\frac{1}{2}$ associated to a transient symmetric Markov jump process on a network is half the square of the Gaussian free field. We construct a coupling between these loops and the free field such that an additional constraint holds: the sign of the free field is constant on each cluster of loops. As a consequence of our coupling we deduce that the loop clusters of parameter $\frac{1}{2}$ do not percolate on periodic lattices. We also construct a coupling between the random interlacement on $\mathbb{Z}^{d}$, $d\geq 3$, introduced by Sznitman, and the Gaussian free field, such that the set of vertices visited by the interlacement is contained in a one-sided level set of the free field. We deduce an inequality between the critical level for the percolation by level sets of the free field and the critical parameter for the percolation of the vacant set of the random interlacement.

Article information

Source
Ann. Probab., Volume 44, Number 3 (2016), 2117-2146.

Dates
Received: May 2014
Revised: March 2015
First available in Project Euclid: 16 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1463410040

Digital Object Identifier
doi:10.1214/15-AOP1019

Mathematical Reviews number (MathSciNet)
MR3502602

Zentralblatt MATH identifier
1348.60141

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G15: Gaussian processes 60G60: Random fields 60J25: Continuous-time Markov processes on general state spaces

Keywords
Gaussian free field loop soup Poisson ensemble of Markov loops percolation by loops random interlacements

Citation

Lupu, Titus. From loop clusters and random interlacements to the free field. Ann. Probab. 44 (2016), no. 3, 2117--2146. doi:10.1214/15-AOP1019. https://projecteuclid.org/euclid.aop/1463410040


Export citation

References

  • [1] Baxter, J. R. and Chacon, R. V. (1984). The equivalence of diffusions on networks to Brownian motion. In Conference in Modern Analysis and Probability (New Haven, Conn., 1982). Contemp. Math. 26 33–48. Amer. Math. Soc., Providence, RI.
  • [2] Bricmont, J., Lebowitz, J. L. and Maes, C. (1987). Percolation in strongly correlated systems: The massless Gaussian field. J. Stat. Phys. 48 1249–1268.
  • [3] Chang, Y. and Sapozhnikov, A. (2014). Phase transition in loop percolation. Preprint. Available at arXiv:1403.5687.
  • [4] Dynkin, E. B. (1984). Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55 344–376.
  • [5] Dynkin, E. B. (1984). Local times and quantum fields. In Seminar on Stochastic Processes, 1983 (Gainesville, Fla., 1983). Progr. Probab. Statist. 7 69–83. Birkhäuser, Boston, MA.
  • [6] Enriquez, N. and Kifer, Y. (2001). Markov chains on graphs and Brownian motion. J. Theoret. Probab. 14 495–510.
  • [7] Fitzsimmons, P. J. and Rosen, J. S. (2014). Markovian loop soups: Permanental processes and isomorphism theorems. Electron. J. Probab. 19 1–30.
  • [8] Folz, M. (2014). Volume growth and stochastic completeness of graphs. Trans. Amer. Math. Soc. 366 2089–2119.
  • [9] Gandolfi, A., Keane, M. S. and Newman, C. M. (1992). Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields 92 511–527.
  • [10] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [11] Häggström, O. and Jonasson, J. (2006). Uniqueness and non-uniqueness in percolation theory. Probab. Surv. 3 289–344.
  • [12] Janson, S. (1984). Bounds on the distributions of extremal values of a scanning process. Stochastic Process. Appl. 18 313–328.
  • [13] Le Jan, Y. (2011). Markov Paths, Loops and Fields. Lecture Notes in Math. 2026. Springer, Heidelberg.
  • [14] Le Jan, Y. and Lemaire, S. (2013). Markovian loop clusters on graphs. Illinois J. Math. 57 525–558.
  • [15] Le Jan, Y., Marcus, M. B. and Rosen, J. (2015). Permanental fields, loop soups and continuous additive functionals. Ann. Probab. 43 44–84.
  • [16] Lupu, T. (2013). Poissonian ensembles of loops of one-dimensional diffusions. Preprint. Available at arXiv:1302.3773.
  • [17] Marcus, M. B. and Rosen, J. (2006). Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics 100. Cambridge Univ. Press, Cambridge.
  • [18] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [19] Rodriguez, P.-F. and Sznitman, A.-S. (2013). Phase transition and level-set percolation for the Gaussian free field. Comm. Math. Phys. 320 571–601.
  • [20] Rozanov, Yu. A. (1982). Markov Random Fields. Springer, New York.
  • [21] Sidoravicius, V. and Sznitman, A.-S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831–858.
  • [22] Sznitman, A.-S. (2010). Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 2039–2087.
  • [23] Sznitman, A.-S. (2012). An isomorphism theorem for random interlacements. Electron. Commun. Probab. 17 1–9.