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Hiking on the Gaussian free field with an altimeter and a compass

Mathematical Physics & Probability

Speaker: Scott Sheffield, UC Berkeley
Location: 693 Kerr
Start time: Tue, Oct 5 2004, 3:10PM

A hiker holding an analog altimeter (with a needle indicating altitude modulo a constant $\chi$) in one hand and a compass in the other traces an altimeter-compass ray (ACR) by walking in such a way that the altimeter and compass needles are always aligned (and thus direction is a linear function of altitude). Formally, if the terrain is described by a smooth real-valued function $h$ on a planar domain, an ACR is a flow line of one of the complex vector fields $e^{2\pi i (\alpha + h/\chi)}$ (where $\alpha \in [0, 1)$ depends on how the hiker holds the altimeter). When $h$ is constant, the ACRs are the rays of Euclidean geometry. We show how to construct contour lines and ACRs when $h$ is a certain wildly fluctuating random distribution called the Gaussian free field. In this case, the ACRs are random fractal paths whose Hausdorff dimensiondepends on $\chi$, and the contour lines of $h$ are random fractal loops with dimension $3/2$. We relate these random collections of fractal rays and loops to the celebrated Schramm-Loewner evolution and use them to describe solutions to some longstanding open problems from statistical physics. This talk is based on joint work with Oded Schramm.