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Decomposition of Brownian loop-soup clusters. (English) Zbl 07117731
Summary: We study the structure of Brownian loop-soup clusters in two dimensions. Among other things, we obtain the following decomposition of the clusters with critical intensity: If one conditions a loop-soup cluster on its outer boundary \(\partial\) (which is known to be an \(\text{SLE}_4\)-type loop), then the union of all excursions away from \(\partial\) by all the Brownian loops in the loop-soup that touch \(\partial\) is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by \(\partial\).
A related result that we derive and use is that the couplings of the Gaussian Free Field (GFF) with \(\text{CLE}_4\) via level lines (by Miller-Sheffield), of the square of the GFF with loop-soups via occupation times (by Le Jan), and of the \(\text{CLE}_4\) with loop-soups via loop-soup clusters (by Sheffield and Werner) can be made to coincide. An instrumental role in our proof of this fact is played by Lupu’s description of \(\text{CLE}_4\) as limits of discrete loop-soup clusters.

MSC:
60J65 Brownian motion
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60G60 Random fields
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[1] Aru, J., Sepulveda, A., Werner, W.: On bounded-type thin local sets (BTLS) of the twodimensional Gaussian free field. J. Inst. Math. Jussieu 18, 591-618 (2019)Zbl 07051731 MR 3936643 · Zbl 07051731
[2] Beffara, V.: The dimension of the SLE curves. Ann. Probab. 36, 1421-1452 (2008) Zbl 1165.60007 MR 2435854 · Zbl 1165.60007
[3] Burdzy, K., Lawler, G. F.: Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal. Ann. Probab. 18, 981-1009 (1990)Zbl 0719.60085 MR 1062056 · Zbl 0719.60085
[4] Camia, F., Lis, M.: Non-backtracking loop soups and statistical mechanics on spin networks. Ann. Henri Poincar´e 18, 403-433 (2017)Zbl 1366.82020 MR 3596767 · Zbl 1366.82020
[5] Dub´edat, J.: SLE and the free field: partition functions and couplings. J. Amer. Math. Soc. 22, 995-1054 (2009)Zbl 1204.60079 MR 2525778 · Zbl 1204.60079
[6] Kemppainen, A., Werner, W.: The nested conformal loop ensembles in the Riemann sphere. Probab. Theory Related Fields 165, 835-866 (2016)Zbl 1352.60117 MR 3520020 · Zbl 1352.60117
[7] Lawler, G. F., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Amer. Math. Soc. 16, 917-955 (2003)Zbl 1030.60096 MR 1992830 · Zbl 1030.60096
[8] Lawler, G. F., Schramm, O., Werner, W.: Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187, 275-308 (2001)Zbl 0993.60083 MR 1879851 · Zbl 0993.60083
[9] Lawler, G. F., Trujillo-Ferreras, J.: Random walk loop soup. Trans. Amer. Math. Soc. 359, 767-787 (2007)Zbl 1120.60037 MR 2255196 · Zbl 1120.60037
[10] Lawler, G. F., Werner, W.: Intersection exponents for planar Brownian motion. Ann. Probab. 27, 1601-1642 (1999)Zbl 0965.60071 MR 1742883 · Zbl 0965.60071
[11] Lawler, G. F., Werner, W.: Universality for conformally invariant intersection exponents. J. Eur. Math. Soc. 2, 291-328 (2000)Zbl 1098.60081 MR 1796962 · Zbl 1098.60081
[12] Lawler, G. F., Werner, W.: The Brownian loop soup. Probab. Theory Related Fields 128, 565-588 (2004)Zbl 1049.60072 MR 2045953 · Zbl 1049.60072
[13] Le Gall, J.-F.: Some properties of planar Brownian motion. In: Cours de l’´ecole de probabilit´es de St-Flour XX, Lecture Notes in Math. 1527, Springer, 111-229 (1993) Zbl 0779.60068 MR 1229519
[14] Le Jan, Y.: Markov Paths, Loops and Fields. Cours de l’´ecole de probabilit´es de St-Flour XXXVIII, Lecture Notes in Math. 2026, Springer (2011)Zbl 1231.60002 MR 2815763
[15] Lupu, T.: From loop clusters and random interlacements to the free field. Ann. Probab. 44, 2117-2146 (2016)Zbl 1348.60141 MR 3502602 · Zbl 1348.60141
[16] Lupu, T.: Convergence of the two-dimensional random walk loop soup clusters to CLE. J. Eur. Math. Soc. 21, 1201-1227 (2019)Zbl 07047497 MR 3941462 · Zbl 07047497
[17] Lupu, T., Werner, W.: A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field. Electron. Comm. Probab. 21, art. 13, 7 pp. (2016)Zbl 1338.60236 MR 3485382 · Zbl 1338.60236
[18] Miller, J. P., Sheffield, S.: Private communication (2010)
[19] Miller, J. P., Sheffield, S.: Imaginary geometry I. Interacting SLEs. Probab. Theory Related Fields 64, 553-705 (2016)Zbl 1336.60162 MR 3477777 · Zbl 1336.60162
[20] Miller, J. P., Sheffield, S.: Imaginary geometry II. Reversibility of SLEκ(ρ1;ρ2)for κ ∈ (0, 4), Ann. Probab. 44, 1647-1722 (2016)Zbl 1344.60078 MR 3502592 Decomposition of Brownian loop-soup clusters3253 · Zbl 1344.60078
[21] Miller, J. P., Sheffield, S.: Imaginary geometry III. Reversibility of SLEκfor κ ∈ (4, 8). Ann. of Math. 184, 455-486 (2016)Zbl 1393.60092 MR 3548530 · Zbl 1393.60092
[22] Miller, J. P., Sheffield, S., Werner, W.: CLE percolations. Forum Math. Pi 5, art. e4, 102 pp. (2017)Zbl 1390.60356 MR 3708206 · Zbl 1390.60356
[23] Miller, J. P., Watson, S. S., Wilson, D. B.: Extreme nesting in the conformal loop ensemble. Ann. Probab. 44, 1013-1052 (2016)Zbl 1347.60061 MR 3474466 · Zbl 1347.60061
[24] Nacu, S¸., Werner, W.: Random soups, carpets and fractal dimensions. J. London Math. Soc. 83, 789-809 (2011)Zbl 1223.28012 MR 2802511 · Zbl 1223.28012
[25] Rohde, S., Schramm, O.: Basic properties of SLE. Ann. of Math. 161, 883-924 (2005) Zbl 1081.60069 MR 2153402 · Zbl 1081.60069
[26] Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221-288 (2000)Zbl 0968.60093 MR 1776084 · Zbl 0968.60093
[27] Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202, 21-137 (2009)Zbl 1210.60051 MR 2486487 · Zbl 1210.60051
[28] Schramm, O., Sheffield, S.: A contour line of the continuous Gaussian free field. Probab. Theory Related Fields 157, 47-80 (2013)Zbl 1331.60090 MR 3101840 · Zbl 1331.60090
[29] Schramm, O., Sheffield, S., Wilson, D. B.: Conformal radii for conformal loop ensembles. Comm. Math. Phys. 288, 43-53 (2009)Zbl 1331.60090 MR 2491617 · Zbl 1187.82044
[30] Sheffield, S.: Exploration trees and conformal loop ensembles. Duke Math. J. 147, 79-129 (2009)Zbl 1170.60008 MR 2494457 · Zbl 1170.60008
[31] Sheffield, S., Watson, S. S., Wu, H.: In preparation
[32] Sheffield, S., Werner, W.: Conformal loop ensembles: The Markovian characterization and the loop-soup construction. Ann. of Math. 176, 1827-1917 (2012)Zbl 1271.60090 MR 2979861 · Zbl 1271.60090
[33] Sznitman, A.-S.: An isomorphism theorem for random interlacements, Electron. Comm. Probab. 17, art. 9, 9 pp. (2012)Zbl 1247.60135 MR 2892408
[34] Sznitman, A.-S.: On scaling limits and Brownian interlacements. Bull. Braz. Math. Soc. 44, 555-592 (2013)Zbl 1303.60022 MR 3167123 · Zbl 1303.60022
[35] Werner, W.: SLEs as boundaries of clusters of Brownian loops. C. R. Math. Acad. Sci. Paris 337, 481-486 (2003)Zbl 1029.60085 MR 2023758 · Zbl 1029.60085
[36] Werner, W.: Conformal restriction and related questions. Probab. Surveys 2, 145-190 (2005) Zbl 1189.60032 MR 2178043
[37] Werner, W.: Some recent aspects of random conformally invariant systems. In: Mathematical Statistical Physics, Elsevier, Amsterdam, 57-99 (2006)Zbl 1370.60142 MR 2581883
[38] Werner, W.: The conformally invariant measure on self-avoiding loops. J. Amer. Math. Soc. 21, 137-169 (2008)Zbl 1130.60016 MR 2350053
[39] Werner, W.: Topics on the Gaussian Free Field and CLE(4). Lecture notes (2015)
[40] Werner, W.: On the spatial Markov property of soups of oriented and unoriented loops. In: S´eminaire de Probabilit´es XLVIII, Lecture Notes in Math. 2168, Springer, 481-503 (2016) Zbl 1370.60192 MR 3618142 · Zbl 1370.60192
[41] Werner, W., Wu, H.: On conformally invariant CLE explorations. Comm. Math. Phys. 320, 637-661 (2013)Zbl 1290.60082 MR 3057185 · Zbl 1290.60082
[42] Werner, W., Wu, H.: From CLE(κ) to SLE(κ, ρ). Electron. J. Probab. 18, art. 36, 20 pp. (2013) Zbl 1338.60205 MR 3035764
[43] Wilson, D. B.: Private communication (2014)
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