Competition in strongly correlated electron systems and associated models

In the study of strongly correlated electron systems, many materials and the models used to represent their essential physics display competition between different electronic phases. Experimentally, materials such as $Ba1-xKxBiO3$ and $La2-xSrxCuO4$ amongst many others, have been found to exhibit competition between insulating and metallic behavior, atiferromagnetism and superconductivity, striped-domain wall structures and homogeneous phases, as well as Peierls charge density wave order and superconductivity.; Simulation techniques for using some of these models, specifically the two dimensional attractive and repulsive Hubbard model on both ladders and square lattices, along with the Holstein model, are reviewed. To test the techniques, determinantal quantum Monte Carlo and density matrix renormalization group results are presented and compared. We also generalize the Ferrenberg-Swendsen analysis for classical systems for use on quantum systems. A demonstration of this generalized Ferrenberg-Swendsen technique is then used to show how one can substantially improve the analysis of quantum Monte Carlo simulations.; With these techniques, properties and Green's functions of the models are extracted and explained. Of particular interest is the electron-phonon vertex of the two dimensional Holstein model, which we discover grows rapidly at a phonon wave vector of ($p,p$) and Matsubara frequency of $wm$ on the Fermi surface due to the onset of the Peierls charge density wave order.