Phase diagrams and Fisher zeros for polymer adsorption
Self-avoiding walks (SAWs) are a well-known model for polymers in dilute solution, and when equipped with appropriate Boltzmann weights they can be used to model phenomena like adsorption at an impenetrable surface. However, SAWs are not exactly solvable and there is much that has not been proved rigorously. In light of this, simpler solvable models can be studied, and these often share qualitative (and sometimes, quantitative) properties with the more general model.
I will discuss two recent papers in this direction. The first is on adsorbing and desorbing ring polymers, and attempts to find a solvable analogue of a model studied by Guttmann, Janse van Rensburg, Jensen and Whittington, who conjectured a phase diagram with four distinct phases but were unable to prove this. The second (joint with E. J. Janse van Rensburg) is on the Fisher zeros for adsorbing polymers, where some careful asymptotic analysis can be used to find the limiting curve and even the behaviour of individual zeros as the polymer length increases.
Dr Nick Beaton, The University of Melbourne