zbMATH — the first resource for mathematics

Convergence of the two-dimensional random walk loop-soup clusters to CLE. (English) Zbl 07047497
Summary: We consider the random walk loop-soup of subcritical intensity parameter on the discrete half-plane \(\mathtt{H}:=\mathbb{Z}\times\mathbb{N}\). We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is a \(\mathrm{CLE}_{\kappa}\) conformal loop ensemble.

60G15 Gaussian processes
60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
PDF BibTeX Cite
Full Text: DOI arXiv
[1] Aru, J., Sep´ulveda, A., Werner, W.: On bounded-type thin local sets of the two-dimensional Gaussian free field. J. Inst. Math. Jussieu (online, 2017), 28 pp.
[2] Van de Brug, T., Camia, F., Lis, M.: Random walk loop soups and conformal loop ensembles. Probab. Theory Related Fields 166, 553–584 (2016)Zbl 1357.60049 MR 3547746 · Zbl 1357.60049
[3] Lawler, G. F.: Partition functions, loop measure, and versions of SLE. J. Statist. Phys. 134, 813–837 (2009)Zbl 1168.82006 MR 2518970 · Zbl 1168.82006
[4] Lawler, G. F., Limic, V.: Random Walk: A Modern Introduction, Cambridge Stud. Adv. Math. 123, Cambridge Univ. Press (2010)Zbl 1210.60002 MR 2677157 Convergence of the two-dimensional random walk loop-soup clusters to CLE1227
[5] Lawler, G. F., Trujillo Ferreras, J. A.: Random walk loop soup. Trans. Amer. Math. Soc. 359, 767–787 (2007)Zbl 1120.60037 MR 2255196 · Zbl 1120.60037
[6] Lawler, G. F., Werner, W.: The Brownian loop-soup. Probab. Theory Related Fields 128, 565– 588 (2004)Zbl 1049.60072 MR 2045953
[7] Le Jan, Y.: Markov Paths, Loops and Fields. Lecture Notes in Math. 2026, Springer (2011) Zbl 1231.60002 MR 3640896
[8] Le Jan, Y., Marcus, M. B., Rosen, J.: Permanental fields, loop soups and continuous additive functionals. Ann. Probab. 43, 44–84 (2015)Zbl 1316.60075 MR 3298468 · Zbl 1316.60075
[9] 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
[10] Lupu, T.: Loop percolation on discrete half-plane. Electron. Comm. Probab. 21, no. 30, 9 pp. (2016)Zbl 1338.60235 MR 3485399
[11] Miller, J., Sheffield, S.: CLE(4) and the Gaussian free field. In preparation
[12] 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
[13] Schramm, O., Sheffield, S.: A contour line of the continuum Gaussian free field. Probab. Theory Related Fields 157, 47–80 (2013)Zbl 1331.60090 MR 3101840 · Zbl 1331.60090
[14] 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
[15] Wang, M., Wu, H.: Level lines of Gaussian free field I: zero-boundary GFF. Stochastic Process. Appl. 127, 1045–1124 (2017)Zbl 1358.60066 MR 3619265 · Zbl 1358.60066
[16] Werner, W.: SLEs as boundaries of clusters of Brownian loops. C. R. Math. Acad. Sci. Paris 337, 481–486 (2003)Zbl 1029.60085 MR 3618142 · Zbl 1029.60085
[17] Werner, W.: Random planar curves and Schramm–Loewner evolutions. In: Lectures on Probability Theory and Statistics, Lecture Notes in Math. 1840, Springer, 107–195 (2004) Zbl 1057.60078 MR 2079672 · Zbl 1057.60078
[18] Werner, W.: Conformal restriction and related questions. Probab. Surv. 2, 145–190 (2005) Zbl 1189.60032 MR 2178043
[19] Werner, W., Wu, H.: From CLE(κ) to SLE(κ, ρ)’s. Electron. J. Probab. 18, no. 36, 20 pp. (2013)Zbl 1338.60205 MR 3035764
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.