Higher-order Airy scaling in deformed Dyck paths

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.