Area-width scaling in generalised Motzkin paths

Abstract

We consider a generalised version of Motzkin paths, where horizontal steps have length $\ell$, with $\ell$ being a fixed positive integer. We first give the general functional equation for the area-length generating function of this model. Using a heuristic ansatz, we derive the area-length scaling behaviour in terms of a scaling function in one variable for the special cases of Dyck, (standard) Motzkin and Schr\"oder paths, before generalising our approach to arbitrary $\ell$. We then derive an expression for the generating function of Schr\"oder paths and analyse the scaling behaviour of this function rigorously in the vicinity of the tri-critical point of the model by applying the method of steepest descents for the case of two coalescing saddle points. Our results show that for Dyck and Schr\"oder paths, the heuristic scaling ansatz reproduces the rigorous results.Comment: 20 pages, 3 figure

Higher-Order Airy Scaling in Deformed Dyck Paths

“…The special case w = 0 was analysed in [8]. In Table 1, we summarize the values of the critical exponents for w = − 1 9 and w > − 1 9 . Figure 4 shows a schematic picture of the phase diagram for fixed jump weight w ≥ − 1 9 .…”

“…9 in [6]). By applying the method of steepest descents to the exact solution for the generating function, the same type of scaling behaviour has been proven rigorously to hold for staircase polygons and Dyck and Schröder paths [7,8,9]. These models can be interpreted as directed subclasses of SAP, with a lower sawtooth boundary in the latter two cases.…”