Abstract
We consider continuous time interlacements on ℂd, d ≥ 3, and investigate the scaling limit of their occupation times. In a suitable regime, referred to as the constant intensity regime, this brings Brownian interlacements on ℝd into play, whereas in the high intensity regime the Gaussian free field shows up instead. We also investigate the scaling limit of the isomorphism theorem of [40]. As a by-product, when d = 3, we obtain an isomorphism theorem for Brownian interlacements.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Belius. Cover times in the discrete cylinder. Available at arXiv:1103.2079.
D. Belius. Gumbel fluctuations for cover times in the discrete torus. To appear in “Probab. Theory Relat. Fields”, also available at arXiv:1202.0190.
P. Billingsley. Convergence of probabilitymeasures. Wiley, New York (1968).
K. Burdzy. Multidimensional Brownian Excursions and Potential Theory. Wiley, New York (1987).
J. Černý and S. Popov. On the internal distance in the interlacement set. Electron. J. Probab., 17(29) (2012), 1–25.
J. Černý and A. Teixeira. From random walk trajectories to random interlacements. Ensaios Matemáticos, 23 (2012), 1–78.
J. Černý, A. Teixeira and D. Windisch. Giant vacant component left by a random walk in a random d-regular graph. Ann. Inst. Henri Poincaré Probab. Stat., 47(4) (2011), 929–968.
M. Cranston and T.R. McConnell. The lifetime of conditioned Brownian motion. Z. für Wahrsch. verw. Geb., 65(1) (1983), 1–11.
A. Drewitz, B. Ráth and A. Sapozhnikov. Local percolative properties of the vacant set of random interlacements with small intensity. To appear in “Annales de l’Institut Henri Poincaré, Probabilités et Statistiques”, also available at arXiv:1206.6635.
R. Durrett. Brownian motion and martingales in analysis. Wadsworth, Belmont CA (1984).
A. Dvoretzky, P. Erdös and S. Kakutani. Double points of paths of Brownian motion in n-space. Acta Sci. Math., Szeged, 12B (1950), 75–81.
X. Fernique. Processus linéaires, processus généralisés. Ann. Inst. Fourier, Grenoble, 17(1) (1987), 1–92.
I.M. Gel’fand and N.Ya. Vilenkin. Generalized Functions. Academic Press, New York and London (1964).
R.K. Getoor. Splitting times and shift functionals. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 47 (1979), 69–81.
J. Glimm and A. Jaffe. Quantum Physics. Springer, Berlin (1981).
G.A. Hunt. Markoff chains and Martin boundaries. Illinois J. Math., 4 (1960), 313–340.
K. Itô. Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. Soc. Indust. Appl. Math., Philadelphia (1984).
S. Janson. Gaussian Hilbert Spaces. Cambridge University Press (1997).
O. Kallenberg. Random measures. Academic Press, New York (1976).
H. Lacoin and J. Tykesson. On the easiest way to connect k points in the random interlacements process. Available at arXiv:1206.4216.
G.F. Lawler. Intersections of random walks. Birkhäuser, Basel (1991).
Y. Le Jan. Markov loops and renormalization. Ann. Probab., 38(3) (2010), 1280–1319.
Y. Le Jan. Markov paths, loops and fields, volume 2026 of “Lecture Notes in Math”. Ecole d’Eté de Probabilités de St. Flour, Springer, Berlin (2012).
M.B. Marcus and J. Rosen. Markov processes, Gaussian processes, and local times. Cambridge University Press (2006).
G. Matheron. Random Sets and Integral Geometry. Wiley, New York (1975).
P.-A. Meyer. Théorème de continuité de P. Lévy sur les espaces nucléaires. Séminaire Bourbaki, 311 (1965/66), 509–522.
S. Port and C. Stone. Brownian motion and classical Potential Theory. Academic Press, New York (1978).
E.B. Procaccia and J. Tykesson. Geometry of the random interlacement. Electron. Commun. Probab., 16 (2011), 528–544.
B. Ráth and A. Sapozhnikov. Connectivity properties of random interlacement and intersection of random walks. ALEA Lat. Am. J. Probab. Math. Stat., 9 (2012), 67–83.
B. Ráth and A. Sapozhnikov. The effect of small quenched noise on connectivity properties of random interlacements. Electron. J. Probab., 8(4) (2013), 1–20.
V. Sidoravicius and A.S. Sznitman. Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math., 62(6) (2009), 831–858.
M.L. Silverstein. Symmetric Markov Processes. Lecture Notes in Math., 426 (1974), Springer, Berlin.
B. Simon. The P(φ)2 Euclidean (Quantum) field theory. Princeton University Press (1974).
B. Simon. Functional Integration and Quantum Physics. Academic Press, New York (1979).
B. Simon. Trace Ideals and Their Applications. Am. Math. Soc., Providence, second edition (2005).
A.S. Sznitman. Brownian motion, obstacles and random media. Springer, Berlin (1998).
A.S. Sznitman. On thedomination of random walk on a discrete cylinder by random interlacements. Electron. J. Probab., 14 (2009), 1670–1704.
A.S. Sznitman. Vacant set of random interlacements and percolation. Ann. Math., 171 (2010), 2039–2087.
A.S. Sznitman. Random interlacements and the Gaussian free field. Ann. Probab., 40(6) (2012), 2400–2438.
A.S. Sznitman. An isomorphism theorem for random interlacements. Electron. Commun. Probab., 17(9) (2012), 1–9.
A.S. Sznitman. Topics in occupation times and Gaussian free fields. Zurich Lectures in Advanced Mathematics, EMS, Zurich (2012).
A. Teixeira. Interlacement percolation on transient weighted graphs. Electron. J. Probab., 14 (2009), 1604–1627.
A. Teixeira. On the size of a finite vacant cluster of random interlacements with small intensity. Probab. Theory Relat. Fields, 150(3–4) (2011), 529–574.
A. Teixeira and D. Windisch. On the fragmentation of a torus by random walk. Commun. Pure Appl. Math., 64(12) (2011), 1599–1646.
M. Weil. Quasi-processus. Séminaire de Probabilités IV, Lecture Notes in Math., 124 (1970), 217–239, Springer, Berlin.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported in part by the grant ERC-2009-AdG 245728-RWPERCRI.
About this article
Cite this article
Sznitman, AS. On scaling limits and Brownian interlacements. Bull Braz Math Soc, New Series 44, 555–592 (2013). https://doi.org/10.1007/s00574-013-0025-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-013-0025-7