We use a combinatorial approach to study the trajectory of a light ray constrained to Euclidian plane R^2 with random reflecting obstacles placed throughout R^2. For the 2D Lorentz lattice gas (LLG) model we derive an analogue of Russo's formula of increasing events in percolation.
This thesis examines the mixing times for one-dimensional interacting particle systems. We use the coupling method to study the mixing rates for particle systems on the circle which move according to specific permutations e.g., transpositions and 3-cycles.
The topic of statistical mechanics has been studied for over a century, and it is one of the pillars of modern physics. This theory can be applied to the study of the thermodynamic behavior of large systems of interacting particles, in which case it is referred to as equilibrium statistical...
Stein's method initially introduced in 1970 by C. Stein is a powerful technique for bounding the distance between the laws of two real-valued random variables. Stein's method has been used to prove distributional convergence to many standard probability distributions such as normal, multivariate normal, Poisson and Brownian motion approximation. In...
Statistical mechanics studies the probability that a system is in a certain state given one or more constraints which are usually fixed conserved quantities. It is a particularly useful and powerful approach for problems with a large number of degrees of freedom where a complete knowledge of the system is...
The paper reviews percolation and some of its important properties, particularly on the 2-D square lattice. A bilevel lattice is introduced, with a percolation model representing the spread of a forest fire according to characteristics of the forest. It is proven that the value of the laddering probability may determine...
One of the newer and rapidly developing approaches in quantum computing is based on "quantum walks," which are quantum processes on discrete space that evolve in either discrete or continuous time and are characterized by mixing of components at each step. The idea emerged in analogy with the classical random...
Consider a simple Markov process on [0,T]. The occupation times were extensively studied in the case of T increasing to infinity. Here we develop a method of computing the distribution for occupation times when T is small via integral equations and integral transforms.
We examine a discrete-time quantum walk with two-step memory for a particle on a one-
dimensional infinite space. The walk is defined with a four-state memory space analogous to the
two-state coin space commonly used in discrete time quantum walks, and a method is presented
for calculating the time evolution...