José Trujillo Ferreras

José Trujillo Ferreras
Ph.D. (2005) Cornell University

First Position

Research associate at ETH Zurich


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


Gregory Lawler

Research Area:

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.