José Trujillo Ferreras

Research associate at ETH Zurich

The Random Walk Loop Soup and the Expected Area of the Brownian Loop in the Plane

Gregory Lawler

Probability

Abstract: The Brownian loop soup introduced in [BLS] is a Poissonian realization from a $\sigma$-finite measure on unrooted Brownian loops. This measure is one of the important recent developments in a large program for understanding scaling limits in two
dimensions. In this thesis, we present a random walk loop soup and show that it converges to the Brownian loop soup. The type of convergence that we establish is a strong approximation and makes use of a strong approximation result of the type derived by Komlós, Major, and Tusnády. A detailed proof of the variation of their result that we need is included.

We also study the following problem. Let $B_t, 0 \le t \le 1$ be a planar Brownian loop (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 $\C\setminus B[0,1]$. We show that the expected area of this hull is $\pi / 5.$ We also use a result of Yor about the law of the index of a Brownian loop to show that the expected areas of the regions of index (winding number) $n\in\Z\setminus\{0\}$ are $\frac{1}{2\pi n^2}\,$. As a consequence, we find that the expected area of the region of index zero inside the loop is $\pi / 30$. The proof uses the Schramm Loewner Evolution (SLE).

[BLS] Gregory F. Lawler and Wendelin Werner, The Brownian loop soup, Probab. Theory Related Fields 128 (2004), 565–588.

PhD

PhD