José A. Trujillo Ferreras

arXiv2005

The expected area of the filled planar Brownian loop is Pi/5

Let B_t be a planar Brownian loop of time duration 1 (a Brownian motion conditioned so that B_0 = B_1). We consider the compact hull obtained by filling in all the holes, i.e. the complement of the unique unbounded component of R^2\B[0,1]. We show that the expected area of this hull is Pi/5. The proof uses, perhaps not surprisingly, the Schramm Loewner Evolution (SLE). Also, using the result of Yor about the law of the index of a Brownian loop, we show that the expected areas of the regions of non-zero index n equal 1/(2 Pi n^2). As a consequence, we find that the expected area of the region of index zero inside the loop is Pi/30; this value could not be obtained directly using Yor's index description.

math.PR↗

arXiv2004

Random walk loop soup

The Brownian loop soup introduced in Lawler and Werner (2004) is a Poissonian realization from 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.

math.PR↗