Abstract
We derive a large deviation principle for the density profile of occupation times of random interlacements at a fixed level in a large box of \({\mathbb { Z}}^d\), \(d \ge 3\). As an application, we analyze the asymptotic behavior of the probability that atypically high values of the density profile insulate a macroscopic body in a large box. As a step in this program, we obtain a similar large deviation principle for the occupation-time measure of Brownian interlacements at a fixed level in a large box of \({\mathbb { R}}^d\), and we derive a new identity for the Laplace transform of the occupation-time measure, which is based on the analysis of certain Schrödinger semi-groups.
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References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
Bolthausen, E., Deuschel, J.D.: Critical large deviations for Gaussian fields in the phase transition regime. I. Ann. Probab. 21(4), 1876–1920 (1993)
Cerf, R.: Large deviations for three dimensional supercritical percolation. Astérisque 267, Société Mathématique de France (2000)
Černý, J., Teixeira, A., Windisch, D.: Giant vacant component left by a random walk in a random \(d\)-regular graph. Ann. Inst. Henri Poincaré Probab. Stat. 47(4), 929–968 (2011)
Chen, Z.-Q.: Gaugeability and conditional gaugeability. Trans. AMS 354(11), 4639–4679 (2002)
Chen, Z.-Q., Song, R.: General Gauge and conditional Gauge theorems. Ann. Probab. 30(3), 1313–1339 (2002)
Chung, K.L., Zhao, Z.: From Brownian Motion to Schrödinger’s Equation. Springer, New York (1995)
Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Birkhäuser, Basel (1993)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, Berlin (1998)
Deuschel, J.D., Stroock, D.W.: Large Deviations. Academic Press, Boston (1989)
Drewitz, A., Ráth, B., Sapozhnikov, A.: Local percolative properties of the vacant set of random interlacements with small intensity. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, also available at arXiv:1206.6635 (2012)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin (1994)
Glimm, J., Jaffe, A.: Quantum Physics. Springer, Berlin (1981)
Grimmett, G.: Percolation, 2nd edn. Springer, Berlin (1999)
Krengel, U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)
Lawler, G.F.: Intersections of Random Walks. Birkhäuser, Basel (1991)
Li, X., Sznitman, A.S.: A lower bound for disconnection by random interlacements. Electron. J. Probab. 19(17), 1–26 (2014)
Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic Press, New York (1967)
Port, S.C., Stone, C.J.: Brownian Motion and Classical Potential Theory. Academic Press, New York (1978)
Pinsky, R.G.: Positive Harmonic Functions and Diffusion. Cambridge University Press, Cambridge (1995)
Popov, S., Teixeira, A.: Soft local times and decoupling of random interlacements. To appear in J. Eur. Math. Soc. arxiv: 1212.1605
Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987)
Rudin, W.: Functional Analysis. Tata Mc Graw-Hill, New Delhi (1974)
Sidoravicius, V., Sznitman, A.S.: Percolation for the vacant set of random interlacements. Commun. Pure Appl. Math. 62(6), 831–858 (2009)
Sznitman, A.S.: Brownian Motion, Obstacles and Random Media. Springer, Berlin (1998)
Sznitman, A.S.: On the domination of random walk on a discrete cylinder by random interlacements. Electron. J. Probab. 14, 1670–1704 (2009)
Sznitman, A.S.: Vacant set of random interlacements and percolation. Ann. Math. 171, 2039–2087 (2010)
Sznitman, A.S.: Random interlacements and the Gaussian free field. Ann. Probab. 40(6), 2400–2438 (2012)
Sznitman, A.S.: An isomorphism theorem for random interlacements. Electron. Commun. Probab. 17(9), 1–9 (2012)
Sznitman, A.S.: On scaling limits and Brownian interlacements. Bull. Braz. Math. Soc., New Ser. 44(4), 555–592 (2013) (Special issue IMPA 60 years)
Teixeira, A., Windisch, D.: On the fragmentation of a torus by random walk. Commun. Pure Appl. Math. 64(12), 1599–1646 (2011)
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This research was supported in part by the grant ERC-2009-AdG 245728-RWPERCRI.
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Li, X., Sznitman, AS. Large deviations for occupation time profiles of random interlacements. Probab. Theory Relat. Fields 161, 309–350 (2015). https://doi.org/10.1007/s00440-014-0550-3
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DOI: https://doi.org/10.1007/s00440-014-0550-3