[Paper Review] Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration

Stephen Melczer|arXiv (Cornell University)|Sep 15, 2017

Advanced Combinatorial Mathematics26 references9 citations

TL;DR

This thesis develops effective algorithms for asymptotic analysis of multivariate generating functions using analytic combinatorics in several variables (ACSV), focusing on D-finite diagonals of rational functions. It establishes the first complexity results for ACSV under broad conditions and applies the framework to solve open problems in lattice path enumeration, including asymptotics for walks in orthants and quadrant models with weighted steps.

ABSTRACT

The field of analytic combinatorics, which studies the asymptotic behaviour of sequences through analytic properties of their generating functions, has led to the development of deep and powerful tools with applications across mathematics and the natural sciences. In addition to the now classical univariate theory, recent work in the study of analytic combinatorics in several variables (ACSV) has shown how to derive asymptotics for the coefficients of certain D-finite functions represented by diagonals of multivariate rational functions. We give a pedagogical introduction to the methods of ACSV from a computer algebra viewpoint, developing rigorous algorithms and giving the first complexity results in this area under conditions which are broadly satisfied. Furthermore, we give several new applications of ACSV to the enumeration of lattice walks restricted to certain regions. In addition to proving several open conjectures on the asymptotics of such walks, a detailed study of lattice walk models with weighted steps is undertaken.

Motivation & Objective

• To develop rigorous, effective algorithms for asymptotic analysis of multivariate generating functions in analytic combinatorics in several variables (ACSV).
• To establish the first complexity bounds for ACSV under broadly satisfied conditions, particularly for smooth and multiple point cases.
• To resolve open conjectures on the asymptotic behavior of lattice path models restricted to orthants and the quarter plane.
• To extend ACSV methods to models with weighted steps and longer steps beyond the standard short-step framework.
• To unify and generalize the kernel method and diagonal representation techniques for symmetric and almost symmetric lattice path models.

Proposed method

• Adopt a computer algebra perspective to formalize and implement ACSV algorithms, ensuring correctness and complexity analysis.
• Use multivariate residue theory and Laurent expansions to extract asymptotics from rational diagonals, especially at smooth and multiple critical points.
• Apply the kernel method in higher dimensions to derive uniform diagonal expressions for walks with highly symmetric step sets.
• Generalize the kernel method to models with long steps (e.g., steps with |i|,|j| > 1) by constructing birational transformation groups that preserve the kernel.
• Employ resultants and algebraic closure techniques to stabilize transformation groups and derive systems of equations for boundary generating functions.
• Use positive series extraction and diagonal extraction via rational functions to express the full generating function in closed form, enabling asymptotic analysis.

Experimental results

Research questions

• RQ1 What are the computational complexity bounds for ACSV algorithms applied to D-finite multivariate generating functions under generic conditions?
• RQ2 How can the kernel method be extended to lattice path models with long steps (e.g., steps with coordinates of modulus >1) in the quarter plane?
• RQ3 What is the asymptotic behavior of lattice path counts in orthants and the quarter plane for models with weighted steps or high symmetry?
• RQ4 Can rational diagonal representations be systematically derived for non-D-finite lattice path generating functions, and under what conditions?
• RQ5 What are the precise asymptotic growth rates for walks in the quarter plane with symmetric or almost symmetric step sets, especially when the exponent is irrational?

Key findings

• The thesis establishes the first complexity results for ACSV algorithms under broad genericity conditions, providing a theoretical foundation for effective asymptotic analysis.
• It proves that the generating function for walks in the first quadrant using the step set S₂ = {(−1,−1),(0,−1),(0,1),(1,0),(−1,0)} is non-D-finite, as the asymptotic growth exhibits an irrational exponent α.
• For the model with step set S = {(1,0), (−1,0), (−2,1), (0,−1)}, the dominant asymptotic expansion of the total generating function is [tk]Q(1,1,t) = (2√3)^k / k^4 * (Ck + O(1/k)), with Ck depending on parity.
• The asymptotic growth for centrally weighted Gouyou-Beauchamps models is shown to be of the form Ck^α ρ^k with irrational α, confirming non-D-finiteness.
• A uniform diagonal expression is derived for almost highly symmetric lattice path models in higher dimensions, depending only on the characteristic polynomial.
• The framework successfully resolves several open conjectures on asymptotics of lattice walks in orthants and the quarter plane, particularly for symmetric and weighted models.