# zbMATH — the first resource for mathematics

On scaling limits and Brownian interlacements. (English) Zbl 1303.60022
The author investigates the scaling limit of the field of occupation times of continuous time interlacements on $$\mathbb{Z}^d$$, $$d\geq3$$. Let $$\mathcal{L}^N$$, $$N\geq1$$, be the random measures on $${\mathbb{R}^d}$$ given by the equation ${\mathcal{L}^N=\frac{1}{dN^2}\sum_{x\in\mathbb{Z}^d}L_{x, u_N}\delta_{x/N}},$ where $$L_{x, u_N}$$ is the field of occupation times of random interlacements at level $$u_N$$ and $$\{u_N\}_{N\geq1}$$ is a suitably chosen sequence of positive numbers.
The first main result states that, in the constant intensity regime ($$u_N=d\alpha N^{d-2}$$, $$\alpha>0$$), the random measures $${\mathcal{L}^N}$$ converges in distribution to $${\mathcal{L}_\alpha,}$$ as $${N\to\infty},$$ where $${\mathcal{L}_\alpha}$$ denotes the occupation-time measure of Brownian interlacements at level $$\alpha$$.
The second main result states that, in the high intensity regime ($${u_NN^{d-2}\to\infty}$$), convergence in distribution to the massless Gaussian free field holds for $\hat{\mathcal{L}}^N=\sqrt{\frac{d}{2N^{2-d}u_N}}(\mathcal{L}^N-\mathbb{E}[\mathcal{L}^N]).$
At the end of paper, there is the scaling limit theorem considered using the isomorphism theorem due to the author [Electron. Commun. Probab. 17, Paper No. 9, 9 p. (2012; Zbl 1247.60135)] and it is applied to the case $$d=3$$.

##### MSC:
 60F05 Central limit and other weak theorems 60J65 Brownian motion 60G60 Random fields 60J27 Continuous-time Markov processes on discrete state spaces
Full Text:
##### References:
 [1] D. Belius. Cover times in the discrete cylinder. Available at arXiv:1103.2079. · Zbl 1247.60135 [2] D. Belius. Gumbel fluctuations for cover times in the discrete torus. To appear in “Probab. Theory Relat. Fields”, also available at arXiv:1202.0190. · Zbl 1295.60053 [3] P. Billingsley. Convergence of probabilitymeasures. Wiley, New York (1968). · Zbl 0172.21201 [4] K. Burdzy. Multidimensional Brownian Excursions and Potential Theory. Wiley, New York (1987). · Zbl 0691.60066 [5] Černý, J; Popov, S, On the internal distance in the interlacement set, Electron. J. Probab., 17, 1-25, (2012) · Zbl 1245.60090 [6] Černý, J; Teixeira, A, From random walk trajectories to random interlacements, Ensaios Matemáticos, 23, 1-78, (2012) · Zbl 1269.60002 [7] 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, 929-968, (2011) · Zbl 1267.05237 [8] Cranston, M; McConnell, TR, The lifetime of conditioned Brownian motion, Z. für Wahrsch. verw. Geb., 65, 1-11, (1983) · Zbl 0506.60071 [9] 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. [10] R. Durrett. Brownian motion and martingales in analysis. Wadsworth, Belmont CA (1984). · Zbl 0554.60075 [11] Dvoretzky, A; Erdös, P; Kakutani, S, Double points of paths of Brownian motion in n-space, Acta Sci. Math., Szeged, 12B, 75-81, (1950) · Zbl 0036.09001 [12] Fernique, X, Processus linéaires, processus généralisés, Ann. Inst. Fourier, Grenoble, 17, 1-92, (1987) · Zbl 0167.16702 [13] I.M. Gel’fand and N.Ya. Vilenkin. Generalized Functions. Academic Press, New York and London (1964). · Zbl 0115.33101 [14] Getoor, RK, Splitting times and shift functionals, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 47, 69-81, (1979) · Zbl 0394.60073 [15] J. Glimm and A. Jaffe. Quantum Physics. Springer, Berlin (1981). · Zbl 0461.46051 [16] Hunt, GA, Markoff chains and martin boundaries, Illinois J. Math., 4, 313-340, (1960) · Zbl 0094.32103 [17] Itô, K, Foundations of stochastic differential equations in infinite dimensional spaces, (1984) · Zbl 0547.60064 [18] S. Janson. Gaussian Hilbert Spaces. Cambridge University Press (1997). · Zbl 0887.60009 [19] O. Kallenberg. Random measures. Academic Press, New York (1976). · Zbl 0345.60032 [20] H. Lacoin and J. Tykesson. On the easiest way to connect k points in the random interlacements process. Available at arXiv:1206.4216. · Zbl 1277.60177 [21] G.F. Lawler. Intersections of random walks. Birkhäuser, Basel (1991). · Zbl 1228.60004 [22] Jan, Y, Markov loops and renormalization, Ann. Probab., 38, 1280-1319, (2010) · Zbl 1197.60075 [23] 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). · Zbl 1231.60002 [24] Marcus, MB; Rosen, J, Markov processes, Gaussian processes, and local times, (2006) · Zbl 1129.60002 [25] G. Matheron. Random Sets and Integral Geometry. Wiley, New York (1975). · Zbl 0321.60009 [26] P.-A. Meyer. Théorème de continuité de P. Lévy sur les espaces nucléaires. Séminaire Bourbaki, 311 (1965/66), 509-522. [27] S. Port and C. Stone. Brownian motion and classical Potential Theory. Academic Press, New York (1978). · Zbl 0413.60067 [28] Procaccia, EB; Tykesson, J, Geometry of the random interlacement, Electron. Commun. Probab., 16, 528-544, (2011) · Zbl 1254.60018 [29] Ráth, B; Sapozhnikov, A, Connectivity properties of random interlacement and intersection of random walks, ALEA Lat. Am. J. Probab. Math. Stat., 9, 67-83, (2012) · Zbl 1277.60182 [30] Ráth, B; Sapozhnikov, A, The effect of small quenched noise on connectivity properties of random interlacements, Electron. J. Probab., 8, 1-20, (2013) · Zbl 1347.60132 [31] Sidoravicius, V; Sznitman, AS, Percolation for the vacant set of random interlacements, Comm. Pure Appl. Math., 62, 831-858, (2009) · Zbl 1168.60036 [32] Silverstein, ML, Symmetric Markov processes, Lecture Notes in Math., 426, springer, berlin, (1974) [33] B. Simon. The P($$φ$$)_{2}Euclidean (Quantum) field theory. Princeton University Press (1974). · Zbl 1175.81146 [34] B. Simon. Functional Integration and Quantum Physics. Academic Press, New York (1979). · Zbl 0434.28013 [35] B. Simon. Trace Ideals and Their Applications. Am. Math. Soc., Providence, second edition (2005). · Zbl 1074.47001 [36] A.S. Sznitman. Brownian motion, obstacles and random media. Springer, Berlin (1998). · Zbl 0973.60003 [37] A.S. Sznitman. On thedomination of random walk on a discrete cylinder by random interlacements. Electron. J. Probab., 14 (2009), 1670-1704. · Zbl 1196.60170 [38] Sznitman, AS, Vacant set of random interlacements and percolation, Ann. Math., 171, 2039-2087, (2010) · Zbl 1202.60160 [39] Sznitman, AS, Random interlacements and the Gaussian free field, Ann. Probab., 40, 2400-2438, (2012) · Zbl 1261.60095 [40] Sznitman, AS, An isomorphism theorem for random interlacements, Electron. Commun. Probab., 17, 1-9, (2012) · Zbl 1247.60135 [41] A.S. Sznitman. Topics in occupation times and Gaussian free fields. Zurich Lectures in Advanced Mathematics, EMS, Zurich (2012). · Zbl 1246.60003 [42] Teixeira, A, Interlacement percolation on transient weighted graphs, Electron. J. Probab., 14, 1604-1627, (2009) · Zbl 1192.60108 [43] Teixeira, A, On the size of a finite vacant cluster of random interlacements with small intensity, Probab. Theory Relat. Fields, 150, 529-574, (2011) · Zbl 1231.60117 [44] Teixeira, A; Windisch, D, On the fragmentation of a torus by random walk, Commun. Pure Appl. Math., 64, 1599-1646, (2011) · Zbl 1235.60143 [45] Weil, M, Quasi-processus, 217-239, (1970), Berlin · Zbl 0211.21203
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.