UC Berkeley

In the first part of this dissertation, we analyze the eigenvalues of the adjacency matrices of a wide variety of random trees as the size of the trees gets larger.

We show that the empirical spectral distributions for many random tree models converge to a deterministic (model dependent) limit as the number of vertices goes to infinity. Our proof uses a suitable notion of local weak convergence for an ensemble of random trees which is known as probability fringe convergence. We conclude

for ensembles such as the linear preferential attachment models, random recursive trees, and the uniform random trees that the limiting spectral distribution has a set of atoms that is dense in the real line. We employ a simplified version of Karp-Sipser algorithm to obtain lower bounds on the mass assigned to zero by the empirical spectral measures. For the the linear preferential attachment model with parameter a > -1, we show that for any fixed k, the k largest eigenvalues jointly converge in distribution to a non-trivial limit when suitably rescaled.

A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost

surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact. In the second part of this dissertation, we study the set-valued coalescing processes when the underlying process is not Brownian motion on the real line but is one of the following two examples of self-similar processes: Brownian motions on the Sierpinski gasket and stable processes on the real line with stable index greater than one. We show that Arratia's conclusion is still valid for these two examples.

Finally in the third and last part of this dissertation we present a new definition of influences of boolean functions in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is equal to the measure of the boundary with respect to uniform enlargement. We prove analogues of the Kahn-Kalai-Linial (KKL) bound and Russo-type formula for the new definition. As a consequence, we establish a sharp threshold phenomenon for monotone increasing events in the product Gaussian space with respect to the mean parameter and give a statistical application of it. We also obtain isoperimetric inequality for the Gaussian measure on R^n and the class of sets invariant under transitive permutation group of the coordinates.