Discrete Systems in Quantum and Statistical Mechanics

Here we consider operators in the physics literature and explore their discretized counterparts for a better understanding of their behavior. In chapter 2, we discretize a standard Hamiltonian model from quantum mechanics and, in this setting, develop a discretized Gelfand-Yaglom formula. From this discrete set up we are able to develop an alternative regularization for the determinant of a class of operators. We refer to this as the lattice-regularization. In chapters 3 and 4 we develop connections between the asymptotics of the inverse of two different operators: the Kasteleyn operator with interface and the Dirac operator with interface. The definition of these operators are given in their respective chapters. While the former is operator acting on a discrete space and the latter is acting on the plane, there are well established connections between the two. Moreover, in chapter 3, we give a complete picture of the asymptotics across the interface when one half of the lattice is weighted critically and the other half is weighted non-critically