![]() ![]() Here, we show that network topology plays a decisive role in determining the statistics of the emerging field if the underlying lattice is endowed with chiral symmetry. Propagation of coherent light through a disordered network is accompanied by randomization and possible conversion into thermal light. Kondakci, H Esat Abouraddy, Ayman F Saleh, Bahaa E A Lattice topology dictates photon statistics. This work was supported in part by an AT&T graduate fellowship, a University of This is the first algorithm for counting self-avoiding walks in which the error bounds are rigorously controlled. In either case we know we can trust our results and the algorithm is guaranteed to run in polynomial time. Thus our algorithm is reliable, in the sense that it either outputs answers that are guaranteed, with high probability, to be correct, or finds a counterexample to the conjecture. In contrast, we present an efficient algorithm which relies on a single, widely-believed conjecture that is simpler than preceding assumptions and, more importantly, is one which the algorithm itself can test. While there are a number of Monte Carlo algorithms used to count self -avoiding walks in practice, these are heuristic and their correctness relies on unproven conjectures. ![]() This problem arises in the study of the thermodynamics of long polymer chains in dilute solution. The second problem is counting self-avoiding walks in lattices. In addition, we show that these results generalize to counting matchings in any graph which is the Cayley graph of a finite group. The algorithm is based on Monte Carlo simulation of a suitable Markov chain and has rigorously derived performance guarantees that do not rely on any assumptions. We present the first efficient approximation algorithm for computing the number of matchings of any size in any periodic lattice in arbitrary dimension. Fisher, Kasteleyn and Temperley discovered an elegant technique to exactly count the number of perfect matchings in two dimensional lattices, but it is not applicable for matchings of arbitrary size, or in higher dimensional lattices. The first problem arises in the study of the thermodynamical properties of monomers and dimers (diatomic molecules) in crystals. In this thesis we consider two classical combinatorial problems arising in statistical mechanics: counting matchings and self-avoiding walks in lattice graphs. ![]() Counting in Lattices: Combinatorial Problems from Statistical Mechanics. ![]()
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