Higher-order Airy scaling in deformed Dyck paths

TL;DR

This paper introduces a new exactly solvable lattice path model called deformed Dyck paths, which exhibit higher-order multicritical behavior, with a scaling function described by a generalized higher-order Airy integral.

Contribution

It presents the first example of an exactly solvable lattice path model with a higher-order multicritical point and derives its scaling function using advanced asymptotic methods.

Findings

Identifies a higher-order multicritical point in the model

Derives a two-variable scaling function as a generalized Airy integral

Establishes the model as the first of its kind with this property

Abstract

We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, 'jumps' orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with respect to their half-length, area and number of jumps. This represents the first example of an exactly solvable lattice path model showing a higher-order multicritical point. Applying the generalized method of steepest descents, we see that the associated two-variable scaling function is given by the logarithmic derivative of a generalized (higher-order) Airy integral.