## The Annals of Probability

### Random interlacements and the Gaussian free field

Alain-Sol Sznitman

#### Abstract

We consider continuous time random interlacements on ${\mathbb{Z}}^{d}$, $d\ge3$, and characterize the distribution of the corresponding stationary random field of occupation times. When $d=3$, we relate this random field to the two-dimensional Gaussian free field pinned at the origin by looking at scaled differences of occupation times of long rods by random interlacements at appropriately tuned levels. In the main asymptotic regime, a scaling factor appears in the limit, which is independent of the free field, and distributed as the time-marginal of a zero-dimensional Bessel process. For arbitrary $d\ge3$, we also relate the field of occupation times at a level tending to infinity, to the $d$-dimensional Gaussian free field.

#### Article information

Source
Ann. Probab., Volume 40, Number 6 (2012), 2400-2438.

Dates
First available in Project Euclid: 26 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1351258730

Digital Object Identifier
doi:10.1214/11-AOP683

Mathematical Reviews number (MathSciNet)
MR3050507

Zentralblatt MATH identifier
1261.60095

#### Citation

Sznitman, Alain-Sol. Random interlacements and the Gaussian free field. Ann. Probab. 40 (2012), no. 6, 2400--2438. doi:10.1214/11-AOP683. https://projecteuclid.org/euclid.aop/1351258730

#### References

• [1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• [2] Bolthausen, E., Deuschel, J.-D. and Giacomin, G. (2001). Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29 1670–1692.
• [3] Brydges, D., Fröhlich, J. and Spencer, T. (1982). The random walk representation of classical spin systems and correlation inequalities. Comm. Math. Phys. 83 123–150.
• [4] Dembo, A., Peres, Y., Rosen, J. and Zeitouni, O. (2004). Cover times for Brownian motion and random walks in two dimensions. Ann. of Math. (2) 160 433–464.
• [5] Dynkin, E. B. (1983). Markov processes as a tool in field theory. J. Funct. Anal. 50 167–187.
• [6] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
• [7] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston, MA.
• [8] Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge.
• [9] Lawler, G. F. and Werner, W. (2004). The Brownian loop soup. Probab. Theory Related Fields 128 565–588.
• [10] Le Jan, Y. (2010). Markov loops and renormalization. Ann. Probab. 38 1280–1319.
• [11] Le Jan, Y. (2011). Markov Paths, Loops and Fields. Lecture Notes in Math. 2026. Springer, Berlin.
• [12] Lukacs, E. (1970). Characteristic Functions, 2nd ed. Hafner, New York.
• [13] Marcus, M. B. and Rosen, J. (2006). Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics 100. Cambridge Univ. Press, Cambridge.
• [14] Revuz, D. and Yor, M. (1998). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
• [15] Sidoravicius, V. and Sznitman, A.-S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831–858.
• [16] Sidoravicius, V. and Sznitman, A.-S. (2010). Connectivity bounds for the vacant set of random interlacements. Ann. Inst. Henri Poincaré Probab. Stat. 46 976–990.
• [17] Spitzer, F. (1976). Principles of Random Walks, 2nd ed. Graduate Texts in Mathematics 34. Springer, New York.
• [18] Symanzik, K. (1969). Euclidean quantum field theory. In Scuola Internazionale di Fisica “Enrico Fermi”, XLV Corso 152–223. Academic Press, New York.
• [19] Sznitman, A.-S. (2010). Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 2039–2087.
• [20] Sznitman, A. S. (2012). Decoupling inequalities and interlacement percolation on $G\times\mathbb{Z}$. Invent. Math. 187 645–706.
• [21] Teixeira, A. (2009). Interlacement percolation on transient weighted graphs. Electron. J. Probab. 14 1604–1628.
• [22] Teixeira, A. and Windisch, D. (2012). On the fragmentation of a torus by random walk. Comm. Pure Appl. Math. 64 1599–1646.
• [23] Windisch, D. (2008). Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 140–150.