Exact solution of a model of a vesicle attached to a wall subject to   mechanical deformation
  Area-weighted Dyck-paths are a two-dimensional model for vesicles attached to a wall. We model the mechanical response of a vesicle to a pulling force by extending this model.   We obtain an exact solution using two different approaches, leading to a q-deformation of an algebraic functional equation, and a q-deformation of a linear functional equation with a catalytic variable, respectively. While the non-deformed linear functional equation is solved by substitution of special values of the catalytic variable (the so-called "kernel method"), the q-deformed case is solved by iterative substitution of the catalytic variable.   Our model shows a non-trivial phase transition when a pulling force is applied. As soon as the area is weighted with non-unity weight, this transition vanishes. 
Aleksander L. Owczarek and Thomas Prellberg
cond-mat.stat-mech cond-mat.soft math.CO