zbMATH — the first resource for mathematics

Random interlacements and the Gaussian free field. (English) Zbl 1261.60095
From the Introduction: The author considers continuous time random interlacements on \(\mathbb Z^d,\;d\geq 3\), where each doubly infinite trajectory modulo time-shift in the interlacement is decorated by i.i.d. exponential variables with parameter one which specify the time spent by the trajectory at each step. The author is interested in the random field of occupation times, i.e., the total time spent at each site of \(\mathbb Z^d\) by the collection of trajectories with label at most \(u\) in the interlacement point process. When \(d = 3\), the author relates this stationary random field to the two-dimensional Gaussian free field pinned at the origin, by looking at the properly scaled field of differences of occupation times of long rods of size \(N\), when the level \(u\) is either proportional to \(\log N/N\) or much larger than \(\log N/N\). The choice of \(u\) proportional to \(\log N/N\) corresponds to a non-degenerate probability that the interlacement at level \(u\) meets a given rod. In the asymptotic regime, ths brings into play an independent proportionality factor of the Gaussian free field, which is distributed as a certain time-marginal of a zero-dimensional Bessel process. This random factor disappears from the description of the limiting random field, when instead \(uN/\log N\) tends to infinity. For arbitrary \(d\geq 3\), the author also relates the field of occupation times at a level tending to infinity, to the \(d\)-dimensional Gaussian free field.

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J27 Continuous-time Markov processes on discrete state spaces
60F05 Central limit and other weak theorems
Full Text: DOI Euclid arXiv
[1] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[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. · Zbl 1034.82018 · doi:10.1214/aop/1015345767
[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. · doi:10.1007/BF01947075
[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. · Zbl 1068.60018 · doi:10.4007/annals.2004.160.433 · euclid:annm/1111770725
[5] Dynkin, E. B. (1983). Markov processes as a tool in field theory. J. Funct. Anal. 50 167-187. · Zbl 0522.60078 · doi:10.1016/0022-1236(83)90066-6
[6] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes , 2nd ed. North-Holland Mathematical Library 24 . North-Holland, Amsterdam. · Zbl 0684.60040
[7] Lawler, G. F. (1991). Intersections of Random Walks . Birkhäuser, Boston, MA. · Zbl 1228.60004
[8] Lawler, G. F. and Limic, V. (2010). Random Walk : A Modern Introduction. Cambridge Studies in Advanced Mathematics 123 . Cambridge Univ. Press, Cambridge. · Zbl 1210.60002 · doi:10.1017/CBO9780511750854
[9] Lawler, G. F. and Werner, W. (2004). The Brownian loop soup. Probab. Theory Related Fields 128 565-588. · Zbl 1049.60072 · doi:10.1007/s00440-006-0319-6
[10] Le Jan, Y. (2010). Markov loops and renormalization. Ann. Probab. 38 1280-1319. · Zbl 1197.60075 · doi:10.1214/09-AOP509
[11] Le Jan, Y. (2011). Markov Paths , Loops and Fields. Lecture Notes in Math. 2026 . Springer, Berlin. · Zbl 1231.60002 · doi:10.1007/978-3-642-21216-1
[12] Lukacs, E. (1970). Characteristic Functions , 2nd ed. Hafner, New York. · Zbl 0201.20404
[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. · Zbl 1129.60002
[14] Revuz, D. and Yor, M. (1998). Continuous Martingales and Brownian Motion , 3rd ed. Springer, Berlin. · Zbl 1087.60040
[15] Sidoravicius, V. and Sznitman, A.-S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831-858. · Zbl 1168.60036 · doi:10.1002/cpa.20267
[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. · Zbl 1210.60107 · doi:10.1214/09-AIHP335 · eudml:241134
[17] Spitzer, F. (1976). Principles of Random Walks , 2nd ed. Graduate Texts in Mathematics 34 . Springer, New York. · Zbl 0359.60003
[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. · Zbl 1202.60160 · doi:10.4007/annals.2010.171.2039 · annals.princeton.edu
[20] Sznitman, A. S. (2012). Decoupling inequalities and interlacement percolation on \(G\times\mathbb{Z}\). Invent. Math. 187 645-706. · Zbl 1277.60183 · doi:10.1007/s00222-011-0340-9
[21] Teixeira, A. (2009). Interlacement percolation on transient weighted graphs. Electron. J. Probab. 14 1604-1628. · Zbl 1192.60108 · doi:10.1214/EJP.v14-670 · emis:journals/EJP-ECP/_ejpecp/viewarticlef770.html · eudml:228482
[22] Teixeira, A. and Windisch, D. (2012). On the fragmentation of a torus by random walk. Comm. Pure Appl. Math. 64 1599-1646. · Zbl 1235.60143 · doi:10.1002/cpa.20382
[23] Windisch, D. (2008). Random walk on a discrete torus and random interlacements. Electron. Commun. Probab. 13 140-150. · Zbl 1187.60089 · doi:10.1214/ECP.v13-1359 · emis:journals/EJP-ECP/_ejpecp/ECP/viewarticlee04e.html · eudml:231549
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.