Random Schroedinger Operators with Connections to Spectral Properties of Groups and Directed Polymers

This thesis studies random Schroedinger operators with connections to group theory and models from statistical physics. First, we study 1D operators obtained as perturbations of the standard adjacency operator on $\\Z$ by putting random i.i.d. noise with finite logarithmic variance on the edges. We study their expected spectral measures $\\mu_H$ near zero. We prove that the measure exhibits a spike of the form $\\mu_H(-\\varepsilon,\\varepsilon) \\sim \\frac{C}{\\abs{\\log\\varepsilon}^2}$, which was first observed by Dyson for a specific choice of the edge weight distribution. We prove the result in generality, without assuming any regularity of edge weights. We also identify the limiting local eigenvalue distribution, obtained by counting crossings of the Brownian motion derived from the operator. The limiting distribution is different from Poisson and the usual random matrix statistics. The results also hold in the setting where the edge weights are not independent, but are sufficiently ergodic, e.g. exhibit mixing. In conjunction with group theoretic tools, we then use the result to compute Novikov-Shubin invariants, which are group invariants related to the spectral measure, for various groups, including lamplighter groups and lattices in the Lie group Sol. \nSecond, we study similar operators in the two dimensional setting. We construct a random Schroedinger operator on a subset of the hexagonal lattice and study its smallest eigenvalues. Using an asymptotic mapping, we relate these eigenvalues to the partition function of the directed polymer model on the square lattice. For a specific choice of the edge weight distribution, we obtain a model known as the log-Gamma polymer, which is integrable. Recent results about the fluctuations of free energy for the log-Gamma polymer allow us to prove Tracy-Widom type fluctuations for the smallest eigenvalue of the original random Schroedinger operator.

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