Statistical mechanics of directed models of polymers in the square lattice

摘要: Directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical physical sciences literature. These are idealized geometric directed introduced to study phase behaviour polymers, include Dyck paths, partially trees models. closely related studied combinatorics literature (and often exactly solvable). They also simplified versions a number statistical mechanics models, including self-avoiding walk, animals vesicles. The exchange approaches ideas between considerably advanced description understanding this will be explored review. combinatorial nature path makes using generating function most natural. In contrast, approach would introduce partition functions free energies, then investigate these general framework critical phenomena. Generating related. For example, questions regarding limiting energy may approached by considering radius convergence function, scaling properties thermodynamic quantities asymptotic function. review methods for obtaining determining energies linear is presented. decomposition leading functional recursions, as well Temperley method (that implemented creating object, one slice at time). A constant term formulation reviewed. features paths informative about underlying that determine diagrams wider classes polymers. Of particular interest adsorption collapse transitions copolymers. those described tricritical scaling. This reviewed can used apply adsorbing, inflating collapsing paths. Critical exponents variety obtained manner, with it better understanding, classification,