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Description

We consider the spectral distribution of the adjacency matrix for a wide variety of random trees which include, for example, preferential attachment trees, random recursive trees, random binary trees, uniform random trees etc. Using soft arguments, we show that the empirical spectral distribution for a number of different random tree models converges to a non-random (model dependent) distribution. Though it is hard to identify the limiting distributions in general, we have been able to settle some of the questions which arise naturally from the simulations. For example, for the most of the random tree models we consider, the limiting spectral distribution has a set of atoms that is dense in the real line. We obtain precise asymptotics on the mass assigned to zero by the empirical spectral measures via the connection with the cardinality of a maximum matching. For the the linear preferential attachment model with parameter $a > -1$, we show that the suitably rescaled $k$ largest eigenvalues converge jointly. Joint work with Shankar Bhamidi and Steve Evans.