Dalhousie University, June 4 - 7, 2013
In this talk I will describe several recent results concerning the geometry of the free field on fractals or fractal-like graphs. These include:
(i) Maxima of the free field on recurrent fractal graphs. (Kumagai-Zeitouni)
(ii) Entropic repulsion of the free field on high-dimensional Sierpinski carpet graphs. (C.-Ugurcan)
(iii) Regularity properties and level sets of continuous free fields on post-critically finite fractals, with applications to random dendrites. (C., in progress)
If time permits, I will discuss how these results can help answer physically inspired problems on fractals, such as: estimating the time that a random walk covers all vertices of a fractal graph; or determining whether a pinned random interface on a fractal substrate undergoes a wetting transition.
The first half of the talk will introduce some notions of a boundary for this tree and random walk, including the hyperbolic boundary, Martin boundary, and Poisson boundary. Application of these tools in the study of wavelets and analysis on fractals is relatively new, so one aim of this talk it to improve awareness of the tools among the wavelets and fractals community. The second half of the talk will note some connections between characterization theorems for scaling functions (by Hernandez and Weiss) and for low-pass filters (by Gundy) and properties of the Poisson boundary. The results hold for multiresolution analyses in any dimension with any dilation matrix, but due to time constraints will only be stated in the one-dimensional, dilation by $2$ case.