An Infinite Family of Adsorption Models and Restricted Lukasiewicz Paths

Abstract

We define (k,ℓ)-restricted Lukasiewicz paths, k≤ℓ∈ℕ₀, and use these paths as models of polymer adsorption. We write down a polynomial expression satisfied by the generating function for arbitrary values of (k,ℓ). The resulting polynomial is of degree ℓ+1 and hence cannot be solved explicitly for sufficiently large ℓ. We provide two different approaches to obtain the phase diagram. In addition to a more conventional analysis, we also develop a new mathematical characterisation of the phase diagram in terms of the discriminant of the polynomial and a zero of its highest degree coefficient.

We then give a bijection between (k,ℓ)-restricted Lukasiewicz paths and “rise”-restricted Dyck paths, identifying another family of path models which share the same critical behaviour. For (k,ℓ)=(1,∞) we provide a new bijection to Motzkin paths.

We also consider the area-weighted generating function and show that it is a q-deformed algebraic function. We determine the generating function explicitly in particular cases of (k,ℓ)-restricted Lukasiewicz paths, and for (k,ℓ)=(0,∞) we provide a bijection to Dyck paths.