Asymptotic Lattice Path Enumeration Using Diagonals

Abstract

Abstract. This work presents new asymptotic formulas for family of walks in Weyl chambers. The models studied here are defined by step sets which exhibit many symmetries and are restricted to the first orthant. The resulting formulas are very straightforward: the exponential growth of each model is given by the number of steps, while the sub-exponential growth depends only on the dimension of the underlying lattice and the number of steps moving forward in each coordinate. These expressions are derived by analyzing the singular variety of a multivariate rational function whose diagonal counts the lattice paths in question. Additionally, we show how to compute subdominant growth for these models, and how to determine first order asymptotics for excursions.

Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior

“…Note that the power series expansion of 1/H(z, t) has all non-negative coefficients, which will allow us to simplify necessary characterizations of the singularities of G(z, t)/H(z, t) below. In the special case where S is symmetric over all axes, we obtain an expression different from that in [38]; by forcing positivity on our series coefficients we have lost some symmetry and less cancellation occurs. For example, the generating function of the model with all possible steps in 2 dimensions is the diagonal of…”

“…where a i,n counts the number of weighted walks of length n using the steps in S, beginning at the origin, ending at i ∈ N d , and never leaving the non-negative orthant in Z d . Describing a walk of length n ending at i ∈ N d recursively as a walk of length n − 1 followed by a single step, one can show (see Melczer and Mishna [38]) that the generating function satisfies a functional equation of the form…”

“…Early combinatorial works in this area include Kreweras [32] and Gessel [28]; in 2005, Bousquet-Mélou [11] introduced the algebraic kernel method, on which our formal series setup is heavily based, to study a quadrant lattice path model stemming from the work of Kreweras. Gessel and Zeilberger [29] gave representations for lattice path generating functions in so-called Weyl chambers in arbitrary dimension, which are equivalent to the diagonal representations of Melczer and Mishna [38] for the Weyl chamber A d 1 . This has been a fruitful area of research: see also Zeilberger [53], Grabiner and Magyar [30], Tate and Zelditch [51], and Feierl [23,24].…”

“…We now show how to derive several useful expressions for the generating functions of the lattice path models we consider. We closely follow the kernel method as outlined in Bousquet-Mélou and Mishna [13], and a straightforward generalization to higher dimensions discussed by Melczer and Mishna [38]. This approach builds heavily on a probabilistic framework detailed by Fayolle, Iasnogorodski, and Malyshev [21]; see also Bousquet-Mélou and Petkovšek [15] for a more general overview on the kernel method.…”

Weighted lattice walks and universality classes

“…Singularity analysis of generating function expressions has led to new results [18,35], in addition to advances on the work of Denisov and Wachtel [13,16,23]. The techniques of Pemantle and Wilson [43] on analytic combinatorics in several variables, combined with expressions for walk generating functions as rational diagonals, have yielded explicit asymptotics for a large collection of models [36,37], and we take a similar approach to weighted models in this work.…”