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Random walk loop soup. (English) Zbl 1120.60037
Authors’ abstract: The Brownian loop soup introduced by G. F. Lawler and W. Werner [Probab. Theory Relat. Fields 128, No. 4, 565–588 (2004; Zbl 1049.60072)] is a Poissonian realization of a \(\sigma\)-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlós, Major, and Tusnády. To make the paper self-contained, we include a proof of the approximation result that we need.

MSC:
60G15 Gaussian processes
60J65 Brownian motion
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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