Probability models on large random graphs and matrices
- This dissertation studies the asymptotic properties of several probability models on graphs and matrices. In the first part, we consider the well known Ising model on large locally tree-like graphs, and show that in the absence of the external magnetic field, in the low temperature regime, the Ising measure decomposes into plus and minus Ising measure. Additionally, we prove a limit result for the Ising measures, conditioned on the sum of the spins being positive, which provides a deeper insight on the behavior of the Ising measures in the limit. In case the graphs are not locally tree-like, we obtain the mean field behavior in high temperature regime, under a certain criterion. In the second part of the thesis, we study limiting spectral distribution of random matrices. Motivated by a conjecture on the limit law of the empirical spectral distribution of uniformly chosen, oriented d-regular graphs, we consider the sum of d i.i.d Haar distributed unitary/orthogonal matrices, and show that its empirical spectral distribution has rotationally invariant limit law on the complex plane, with a explicit density.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Department of Statistics.
|Statement of responsibility
|Submitted to the Department of Statistics.
|Thesis (Ph.D.)--Stanford University, 2014.
- © 2014 by Anirban Basak
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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