David Croydon

Central limit theorem for the spectrum of a random fractal string

I will discuss the spectral asymptotics of some open subsets of the real line with random fractal boundary. In particular, I will describe a result that ensures the almost-sure existence of the second order term in the asymptotics and gives criteria under which when there will be a central limit theorem that captures the fluctuations around this limit. In a certain case when the random variables governing the fractal have Dirichlet distributions, we are able to exhibit different regimes, depending on the parameters of the distribution, in which there will or will not be a central limit theorem. A central part of the argument is linking the spectrum of the random fractal to a branching process. (To complete our argument, we needed to derive a central limit theorem for such a process under weaker conditions than those previously known.) We expect much of the discussion to be reasonably generic for random self-similar fractals. If time permits, I will also briefly mention an important example of a random fractal tree - the continuum random - to which we can also apply our central limit theorem.
Based on joint work with Phillipe H. A. Charmoy (formerly Oxford) and Ben Hambly (Oxford)