Sanjay Ramgoolam, Ali Zahabi and I have published in J. Physics A (an open access, no author fee journal). The title is Quiver asymptotics: 
free chiral ring. This is a completely new application area for me, and shows how useful the results of the ACSV project are.
Tag Archives: ACSV applications
Lattice walks in the positive orthant
The paper Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior (with the excellent Stephen Melczer) has been accepted by SIAM J. Discrete Math. We consider the enumeration of nearest-neighbor walks on the non-negative lattice in d-dimensional space. Previous work in this area has established asymptotics for the number of walks in certain families of models by applying the techniques of analytic combinatorics in several variables (ACSV), where one encodes the generating function of a lattice path model as the diagonal of a multivariate rational function. Melczer and Mishna obtained asymptotics when the set of steps is symmetric over every axis; in this setting one can always apply the methods of ACSV to a multivariate rational function whose whose set of singularities is a smooth manifold (the simplest case). Here we go further, providing asymptotics for models with generating functions that must be encoded by multivariate rational functions with non-smooth singular sets. In the process, our analysis connects past work to deeper structural results in the theory of analytic combinatorics in several variables. One application is a closed form for asymptotics of models defined by step sets which are symmetric over all but one axis. As a special case, we apply our results in dimension 2 to give a rigorous proof of asymptotics conjectured by Bostan and Kauers; asymptotics for walks returning to boundary axes and the origin are also given.
Twenty combinatorial examples of asymptotics derived from multivariate generating functions
Robin Pemantle and Mark C. Wilson, to appear in SIAM Review, June 2008.
Abstract: Let $latex F$ be a power series in at least two variables that defines a meromorphic function in a neighbourhood of the origin; for example, $latex F$ may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of $latex F$, uniform in certain explicitly defined cones of directions.
The purpose of this article is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The first part reviews the Morse-theoretic underpinnings of these techniques, and then summarizes the necessary results so that only elementary analyses are needed to check hypotheses and carry out computations. The remainder focuses on combinatorial applications. Specific examples deal with enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, descents and solutions to integer equations. After the individual examples, we discuss three broad classes of examples, namely functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. Generating functions derived in these three ways are amenable to our asymptotic analyses, and we state some further general results that apply to these cases.