Friday, October 2, 2020 at 3:30pm to 4:30pm
James Lee, University of Washington
Abstract: A long line of work in anomalous diffusion on fractals and self-similar random media confirms that in the "strongly recurrent" regime, certain spectral and geometric notions satisfy the "Einstein relation" for diffusion. These are relations between scaling exponents that connect the mean displacement and return probabilities of the random walk to the density and conductivity of the underlying medium.
I will show that, in stationary random environments, these relations hold somewhat more generally. In particular, we handle the case of spectral dimension = 2, capturing a rich class of random networks that are not strongly recurrent. This includes many models of random planar networks like the uniform infinite planar triangulation (UIPT).
Our main technical tool is the theory of Markov type and metric embedding theory, combined with a well-chosen change of metric on the underlying graph.
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