Area-width scaling in generalised Motzkin paths
Abstract
We consider a generalised version of Motzkin paths, where horizontal steps have length ℓ, with ℓ being a fixed positive integer. We first give the general functional equation for the area-width generating function of this model. Using a heuristic ansatz, we then derive the area-width scaling behaviour in terms of a scaling function in one variable for the special cases of Dyck, (standard) Motzkin and Schröder paths, before generalising our approach to arbitrary ℓ. We then rigorously derive the tricritical scaling of Schröder paths by applying the generalised method of steepest descents to the known exact solution for their area-width generating function. Our results show that for Dyck and Schröder paths, the heuristic scaling ansatz reproduces the rigorous results.
- Publication:
-
Physica A Statistical Mechanics and its Applications
- Pub Date:
- September 2017
- DOI:
- 10.1016/j.physa.2017.04.151
- arXiv:
- arXiv:1605.09643
- Bibcode:
- 2017PhyA..482..611H
- Keywords:
-
- Exact solutions;
- Scaling functions;
- Basic hypergeometric series;
- Schroeder paths;
- Saddle point method;
- Dominant balance;
- Condensed Matter - Statistical Mechanics;
- Mathematics - Combinatorics
- E-Print:
- 20 pages, 3 figures