Rapid mixing of Swendsen-Wang dynamics in two dimensions

We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs. In particular, we prove that rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum degree and that rapid mixing of SW and rapid mixing of SB are equivalent. Additionally, the spectral gap of SW and SB on planar graphs is bounded from above and from below by the spectral gap of these dynamics on the corresponding dual graph with suitably changed temperature. As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional Ising model at all temperatures. Furthermore, we obtain new results for general graphs at high or low enough temperatures.Comment: Ph.D. thesis, 66 page

Ferromagnetic ground states of the Hubbard model on a complete graph

We use group theory to derive the exact analytical expression of the ferromagnetic ground states of the Hubbard model on a complete graph for arbitrary lattice sites f and for arbitrary fillings $N$. We find that for $t>0$ and for $N=f+1$ the ground state is maximally ferromagnetic with total spin $S=(f-1)/2$. For $N > f+1$ the ground state is still ferromagnetic but becomes degenerate with respect to $S$.Comment: latex fil

The role of the family in educating children for the faith

One of the major responsibilities of the Christian Family is not only taking care of their children but also the education of the children for the faith. This role which is so important today is iterated in many Church documents. This is not a recent addition to the responsibilities of the family; in fact, it has been the responsibility of the family since biblical times. This paper seeks to highlight the importance of the role of the family in the education for the faith of children, starting as far back as biblical times, and reaching our contemporary times by reviewing the most important Church documents. Once the role of the family in educating for the faith of children is established, then some practical ways in which the family can educate for the faith today are discussed.peer-reviewe