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Random walk loop soups and conformal loop ensembles

T. van de Brug ; F. Camia ; Marcin Lis (Institutionen för matematiska vetenskaper, matematisk statistik)
Probability Theory and Related Fields (0178-8051). Vol. 166 (2016), 1, p. 553-584.
[Artikel, refereegranskad vetenskaplig]

The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm-Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, "large" lattice loops from the random walk loop soup converge to "large" loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of "large" lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of "large" lattice loops converges to CLE.

Nyckelord: Brownian loop soup, Random walk loop soup, Planar Brownian motion, Outer boundary, Conformal, critical percolation, convergence, invariance, exponents, plane, Mathematics

Denna post skapades 2016-10-28. Senast ändrad 2016-11-22.
CPL Pubid: 244411


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Institutionen för matematiska vetenskaper, matematisk statistik (2005-2016)



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