Forces and pressures in adsorbing partially directed walks

journal_title: Journal of Physics A: Mathematical and Theoretical article_type: paper article_title: Forces and pressures in adsorbing partially directed walks copyright_information: © 2016 IOP Publishing Ltd date_received: 2015-10-02 date_accepted: 2016-03-18 date_epub: 2016-04-1

Exchange Symmetries in Motzkin Path and Bargraph Models of Copolymer Adsorption

In a previous work [26], by considering paths that are partially weighted, the generating function of Dyck paths was shown to possess a type of symmetry, called an exchange relation, derived from the exchange of a portion of the path between weighted and unweighted halves. This relation is particularly useful in solving for the generating functions of certain models of vertex-coloured Dyck paths; this is a directed model of copolymer adsorption, and in a particular case it is possible to find an asymptotic expression for the adsorption critical point of the model as a function of the colouring. In this paper we examine Motzkin path and partially directed walk models of the same adsorbing directed copolymer problem. These problems are an interesting generalisation of previous results since the colouring can be of either the edges, or the vertices, of the paths. We are able to find..