The Statistical Mechanics and Combinatorics of Self-Avoiding Walk and Directed Path Models of Polymers

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The polymer entropy problem underlies a rich and varied mathematical world, including lattice models such as the self-avoiding walk and related models, directed path models in combinatorial mathematics, percolation, as well as numerical methods including Monte Carlo methods and transfer matrix approaches. The models are ubiquitous in statistical mechanics and in the theory of phase transitions, making this field one which straddles rigorous and applied statistical mechanics, combinatorial mathematics, probability theory, and mathematical physics. There is also a connection to experimental polymer physics because scaling and phase behaviour in the models are related to the physical properties of polymers, for example, the adsorption transition in a polymer can be modelled by an adsorbing self-avoiding walk model. My research program into aspects of the polymer entropy problem relates in particular to the rigorous analysis of lattice models, numerical simulation to examine phase behaviour in the models and to collect data on knot entropy by modelling knotted lattice polygons, and the exact solution and asymptotic analysis of directed lattice path models. I propose to continue my activity in each of these different areas. In numerical work our construction of GARM and GAS Monte Carlo algorithms gave us an efficient way to use microcanonical sampling. This sampling is effective in the knot entropy problem, and I will be continuing to work in this area. My immediate goals are the thermodynamic properties of lattice models of knotted ring polymers. The GARM and GAS algorithms are also efficient at sampling lattice walks and polygons, and we recently computed the entropic pressure field near a square lattice polygon, determining the scaling properties of the pressure field. It is an immediate goal to apply our methods to related models of walks, lattice animals and trees, in each case examining the pressure and its scaling properties. Recent work with collaborators on directed path models enabled us to determine the scaling of the pressure field near a directed lattice path. This work is ongoing, and we are working on more general models, including partially directed paths. These models require ever more sophisticated methods, and the kernel method for solving functional recurrences have proven very useful. My aim is to advance this field by applying the kernel method to more general models, perhaps including interacting directed vesicles, to gain insight in both the mathematical properties, phase diagrams, and scaling of directed models. Techniques such as atomic force microscopy makes it possible to subject adsorbed polymers to pulling forces. We have modelled this by subjected an adsorbing self-avoiding walk to an externally pulled force. We proved existence of a thermodynamic limit in the model, and by examining the asymptotic shape of a phase boundary separating a ballistic phase from an adsorbed phase, proved that the phenomenon of re-entrance occurs in this model. An immediate goal is to extend our results to models with forces pulling parallel or at an angle to the adsorbing surface. Overall my research program straddles areas of numerical work (Monte Carlo methods of discrete lattice objects), with exact combinatorial methods (variants of directed lattice path models and their exact solutions) and with rigorous approaches to interacting models related to the self-avoiding walk. An important additional interest is the knot entropy problem, in particular the entropy of knotted lattice polygons (these are models of knotted polymers, and are often related to knotting in DNA molecules).