Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior
aa r X i v : . [ m a t h . C O ] M a y Higher Dimensional Lattice Walks: Connecting Combinatorialand Analytic Behavior
Stephen Melczer ∗ Mark C. Wilson † Abstract
We consider the enumeration of walks on the non-negative lattice N d , with steps defined by a set S ⊂{− , , } d \{ } . Previous work in this area has established asymptotics for the number of walks in certainfamilies of models by applying the techniques of analytic combinatorics in several variables (ACSV), whereone encodes the generating function of a lattice path model as the diagonal of a multivariate rationalfunction. Melczer and Mishna obtained asymptotics when the set of steps S is symmetric over everyaxis; in this setting one can always apply the methods of ACSV to a multivariate rational function whoseset of singularities is a smooth manifold (the simplest case). Here we go further, providing asymptoticsfor models with generating functions that must be encoded by multivariate rational functions havingnon-smooth singular sets. In the process, our analysis connects past work to deeper structural results inthe theory of analytic combinatorics in several variables. One application is a closed form for asymptoticsof models defined by step sets that are symmetric over all but one axis. As a special case, we apply ourresults when d = 2 to give a rigorous proof of asymptotics conjectured by Bostan and Kauers; asymptoticsfor walks returning to boundary axes and the origin are also given. Subject classification:
Keywords : lattice path enumeration, kernel method, analytic combinatorics, D-finite, generating function
Much modern research in enumeration concerns links between analytic function behaviour and combinatorialmodels. When one has sufficient information about a generating function — for instance, the locations andtypes of its singularities closest to the origin in the complex plane — the theory of analytic combinatoricsin one variable gives, almost automatically in many cases, the asymptotics of the related sequence (see thecompendium text of Flajolet and Sedgewick [25] for further details).When a generating function is given in terms of an expansion of a multivariate power series, however,much less is known. Over the last two decades, several authors have been working towards the developmentof a theory of analytic combinatorics in several variables. This refers to methods for deriving asymptotics ofa sequence b n = [ z n r ] F ( z ) = [ z n · r · · · z n · r d d ] F ( z , . . . , z d ) := a n r ,...,n r d for some fixed vector r ∈ Z d and multi-dimensional sequence ( a i ) i ∈ Z d such that F ( z ) := X i ∈ Z d a i z i is a meromorphic function analytic in a specified domain. The theory, as it has developed, generally consistsof two stages: first, one must find the contributing singularities of F ( z ) , which are the singularities of F ( z ) ∗ University of Pennsylvania, Department of Mathematics, 209 S. 33rd Street, Philadelphia, PA 19104,([email protected]), ). † Department of Computer Science, University of Auckland, Private Bag 92019 Auckland, New Zealand([email protected]), http://mcw.blogs.auckland.ac.nz/ ). asymptoticcontribution of each point and sum the results.In 2002, Pemantle and Wilson [46] derived asymptotics for the power series expansion of a rational func-tion F ( z ) = G ( z ) /H ( z ) admitting a finite number of contributing points, at which the variety of singularitiesis a complex manifold. Two years later [45], they extended this analysis to allow contributing points wherethe variety of singularities is the transverse union of complex manifolds. Modern approaches incorporatetechniques from differential and algebraic geometry, topology, and singularity theory; the interested readeris referred to the text of Pemantle and Wilson [47] or, for a more elementary introduction, the thesis ofMelczer [36]. The enumeration of lattice walks restricted to certain regions is a classical topic in combinatorics, tracing itsroots back hundreds of years to work on what is now known as the ballot problem. In modern times, a largearea of work on this topic centers around using the so-called kernel method to express generating functions oflarge classes of models as positive series extractions or diagonals of multivariate rational functions, provingthat the generating functions are (or are not)
D-finite ; that is, determining whether they satisfy a lineardifferential equation with polynomial coefficients. Although there are a finite number of models with stepsin {± , } that are restricted to a quarter-plane, their combinatorics has been the subject of intense studyin recent years [21, 11, 31, 41, 13, 7, 8, 50, 42, 37, 38, 4, 12, 2, 6, 19]. The models considered in this paperare higher-dimensional generalizations of this two-dimensional setting.A lattice path model is determined by a finite set of steps S ⊂ Z d and a region P ⊂ Z d to whichthe walks of the model are restricted. A model whose set of allowed steps is symmetric over every axis iscalled highly symmetric . Melczer and Mishna [38] combined the kernel method with techniques from analyticcombinatorics in several variables to give asymptotics for the number of integer lattice walks restricted toan orthant (that is, P = N d ) when the set of steps S is a subset of {± , } d \ { } and is highly symmetric.The main result of their paper is the following. Theorem 1 (Melczer and Mishna [38, Theorem 3.4]) . Let
S ⊂ {− , , } d \ { } be a set of steps thatis symmetric over every axis and moves forwards and backwards in each coordinate. Then the number s n of walks of length n taking steps in S , beginning at the origin, and never leaving the positive orthant hasasymptotic expansion s n = |S| n · n − d/ · (cid:18) (cid:16) s (1) · · · s ( d ) (cid:17) − / π − d/ |S| d/ + O (cid:18) n (cid:19) (cid:19) , where s ( k ) denotes the number of steps in S that have k th coordinate . In order to get this result, Melczer and Mishna used the kernel method to derive the expression F ( t ) = ∆ (cid:18) G ( z , t ) H ( z , t ) (cid:19) = ∆ (cid:18) (1 + z ) · · · (1 + z d )1 − t ( z · · · z d ) S ( z ) (cid:19) for the generating function F ( t ) counting the number of walks of a given length defined by the model, where ∆ is the diagonal operator defined in Section 3.2 below. After verification of certain technical conditions,Theorem 1 then follows from the main result of the original paper of Pemantle and Wilson [46]. Thesymmetry condition on the step set S was chosen by Melczer and Mishna precisely because it leads to asmooth singular variety , the set of singular points of G/H , defined by the zero set of H .Although not originally considered by Melczer and Mishna, Theorem 1 can be easily extended [36, Chapter7] to handle positively weighted step sets with weights that are symmetric over every axis. Generalizationsof this work have used multivariate singularity analysis to enumerate weighted walks [17] and walks withstep coordinates having absolute value greater than one [5].2 .2 Our contributions Building on the conference paper of Melczer and Wilson [39], in this work we generalize the results of Melczerand Mishna by giving asymptotics of lattice path models, restricted to the positive orthant, whose set ofallowable steps is symmetric over all but one axis. The formulae we obtain give explicit and fairly simpledescriptions of the exponential rate, leading order and leading coefficient in terms of the basic data of thewalk step set S . The drift of a walk, the vector sum of all the steps in S , plays a crucial role in asymptotics;owing to symmetry, only one coordinate of the drift may be non-zero and we refer to a walk as havingnegative, zero, or positive drift depending on the sign of this coordinate.Additionally, we show how the arguments of Melczer and Mishna’s analysis fit into larger structuretheorems about the singularities of multivariate rational functions and their asymptotic expansions. In thenegative and zero drift cases, substantial extra work is needed because the expected leading order coefficientfor generic problems of this dimension turns out to vanish. Unfortunately, due to this vanishing and otherdegeneracies in integrals that must be asymptotically approximated, we are currently unable to determineasymptotics for the general zero drift case. Such models will be the subject of future study.Furthermore, we provide the first rigorous proofs of the guessed asymptotics of Bostan and Kauers [7] on2D walks restricted to the non-negative quadrant, completing the outline in Melczer and Wilson [39]. Ouranalysis uncovers and explains periodicity of the asymptotic coefficients in the negative drift case, which wasnot noted in [7], and we give asymptotics for the number of walks returning to the boundary axes and theorigin. Around the same time as the conference paper of Melczer and Wilson [39], Bostan et al. [6] rigorouslygave annihilating differential equations for the generating functions of these lattice path models. Using thesedifferential equations they were able to prove some of the guessed asymptotics of Bostan and Kauers [7],however due to issues related to the decidability of asymptotics for coefficients of D-finite functions theywere unable to prove all asymptotics. We discuss the difficulties they faced, and how our results fit into thiscontext, in Section 7.2. An accompanying Maple worksheet verifying our calculations can be found online . Lattice path models restricted to a halfspace have algebraic generating functions that can be explicitlydetermined [27], leading to strong asymptotic results [1]. For this reason, much attention has been devotedto walks in quadrants and related cones such as orthants. Early combinatorial works in this area includeKreweras [32] and Gessel [28]; in 2005, Bousquet-Mélou [11] introduced the algebraic kernel method , onwhich our formal series setup is heavily based, to study a quadrant lattice path model stemming from thework of Kreweras. Gessel and Zeilberger [29] gave representations for lattice path generating functions inso-called Weyl chambers in arbitrary dimension, which are equivalent to the diagonal representations ofMelczer and Mishna [38] for the Weyl chamber A d . This has been a fruitful area of research: see alsoZeilberger [53], Grabiner and Magyar [30], Tate and Zelditch [51], and Feierl [23, 24]. The systematiccombinatorial enumeration of walks in a quadrant was popularized by Bousquet-Mélou and Mishna [13],following work of Petkovšek [48], Bousquet-Mélou and Petkovšek [14], and Mishna [40], among others.Walks in quadrants and orthants have also been long studied from a probabilistic perspective. In oneapproach, developed in part for problems arising in queuing theory, a singularity analysis of solutions tofunctional equations satisfied by lattice path generating functions yields analytic and asymptotic information.The text of Fayolle et al. [21] gives a detailed view on the techniques involved, some of which inspiredBousquet-Mélou’s creation of the algebraic kernel method; see also Malyšev [35] for an early history. Thelattice path models we study have asymptotics of the form C n α ρ n for constants α and ρ , where C is constantor depends only on the periodicity of n . Fayolle and Raschel [22] used these techniques to outline a methodthat, in principle, allows one to calculate the exponential growth ρ for many quadrant models.Another probabilistic approach to lattice path enumeration is to use local limit theorems and/or approx-imate discrete walks by scaling limits such as multidimensional Brownian motion. For a large variety ofstep sets and restricting cones, including orthants, Denisov and Wachtel [18] give techniques for determin-ing the exponential growth ρ and exponent α for the number of walks that begin and end at the origin: https://github.com/smelczer/HigherDimensionalLatticeWalks ρ and α for the number of walks ending anywhere in the restricting cone. Among otherresults, Duraj [20, Example 7] determines ρ and α for quadrant walks ending anywhere in the quadrantwhen the drift has negative coordinates, and Garbit and Raschel [26] give the exponential growth ρ for walksending anywhere in the restricting cone under no restriction on the drift of a model. We note that using theseprobabilistic techniques it is very difficult, if not impossible, to determine the leading asymptotic constant C or to determine higher order asymptotic terms, as our approach provides (in a less general setting).Finally, lattice path enumeration has been studied through the lens of computer algebra. Among themany results in this area we mention: Kauers et al. [31], which proved a longstanding open problem on theenumeration of certain quarter plane walks using creative telescoping techniques; Bostan and Kauers [7],which computationally guessed certain differential equations satisfied by the generating functions of quarterplane models and used this to guess the asymptotics we prove in Section 6; and Bostan et al. [10], whichused the work of Denisov and Wachtel mentioned above to create algorithms explicitly determining theexponential growth ρ and exponent α for quarter plane models.The enumeration of lattice walks is a thriving area of enumerative combinatorics, with too many resultsto explicitly mention here. Those looking for additional resources can investigate the texts of Mohanty [43]and Narayana [44], and the survey of Krattenthaler [3, Chapter 10]. We begin in Section 2 by discussing our main results and some illustrative examples. Section 3 then gives anoverview of the kernel method applied to lattice path enumeration, and shows how it can be used to deriveexpressions for lattice path generating functions that are amenable to the techniques of analytic combinatoricsin several variables. Unlike the previous work of Melczer and Mishna — where complete symmetry of the stepsets under consideration simplified the required manipulations of the kernel method — care must be takenhere when manipulating diagonal and positive sub-series extractions in iterated Laurent series rings. Section 4details the general methods of analytic combinatorics in several variables, and outlines how the asymptoticanalysis will proceed; in Proposition 15 we give an explicit description of the contributing singularitiesfor the models under consideration. We derive asymptotics using this characterization in Section 5. Thiswork divides naturally into three cases: the positive, negative and zero drift, listed in increasing order ofdifficulty (as mentioned above, we do not treat the general zero drift case here; see Section 7.4 for moreinformation). Detailed computations needed are collected in Appendix A. Section 6 proves asymptotics of2D walks restricted to the non-negative quadrant. Section 7 discusses extensions and directions for futureresearch.A summary of results is displayed in Table 1. It has previously been observed [22, 17] that the sub-exponential order term n α appearing in asymptotics for the number of walks in a lattice path model hassome correlation with the drift of its steps. This phenomenon occurs here: each coordinate where the driftis negative corresponds to a contribution of n − / to dominant asymptotics, a positive drift coordinate doesnot effect the order term, and a zero drift coordinate corresponds to an asymptotic contribution of at most n − / (depending on whether or not the walk is highly symmetric).Drift Exists for Exp. rate Order Geometry Theorempositive d ≥ | S | n − ( d − / nonsmooth Theorem 2negative d ≥ < | S | n − − d/ smooth Theorem 4zero (h.s.) d ≥ | S | n − d/ smooth [38, Thm 3.4]zero (not h.s.) d ≥ | S | O ( n / − d/ ) nonsmooth ConjecturalTable 1: Summary of results: exponential rate and leading asymptotic order. Here “h.s." means “highlysymmetric" (step set is symmetric over every axis). 4 Main Results and Examples
In order to simplify equations, we fix a dimension d ∈ N and use the following notation multi-index: z i = z − i ; z = ( z , . . . , z d ); i = ( i , i , . . . , i d ) ∈ Z d ; z i = z i · · · z i d d ; z ˆ k := ( z , . . . , z k − , z k +1 , . . . , z d ) . We consider walks in dimension d defined by a (finite) set S ⊂ {± , } d \ { } of weighted steps, where i ∈ S is given real weight w i ≥ , such that • there exists some step forwards and some step backwards in the direction of each coordinate axis:For all j = 1 , . . . , d there exists i ∈ S with i j = 1 and w i = 0; For all j = 1 , . . . , d there exists i ∈ S with i j = − and w i = 0 . • the weighting w i is symmetric over all axes except one; • each walk is confined to the non-negative orthant N d .The (weighted) characteristic polynomial of S is the Laurent polynomial S ( z ) = X i ∈S w i z i . We may assume without loss of generality that the axis of non-symmetry is z d . In other words, our step set S is such that for each j with ≤ j ≤ d − , S ( z , . . . , z j − , z j , z j +1 , . . . , z d ) = S ( z ) , so we may write S ( z ) = z d A (cid:0) z ˆ d (cid:1) + Q (cid:0) z ˆ d (cid:1) + z d B (cid:0) z ˆ d (cid:1) for Laurent polynomials A, B, and Q that are symmetric in their variables. Note that S ( ) = |S| is the sizeof the step set when each step has weight .Also important is the drift B ( ) − A ( ) of a walk with respect to the d -axis, the weight of steps inthe positive z d direction minus the weight of steps in the negative z d direction. For our models the signof the drift will correspond to different asymptotic regimes; in general the relationship between drift andasymptotics is more nuanced [17, Section 6.4]. Although the models we consider start at the origin, it ispossible to modify our approach with minimal overhead to start at any i ∈ N d . In fact, one can treat thestarting point i as a parameter that will appear only in the leading constant of dominant asymptotics forthe number of walks, and this approach can be used to construct discrete harmonic functions [17]. For ≤ k ≤ d − , define b k to be the total weight of the steps moving forwards (or backwards) in the k thcoordinate, b k = X i ∈S ,i k =1 w i = X i ∈S ,i k = − w i . Our main asymptotic result for positive drift models is the following.
Theorem 2 (Positive Drift Asymptotics) . Let S be a (weighted) step set that is symmetric over all but oneaxis and takes a step forwards and backwards in each coordinate. If S has positive drift, then then numberof walks of length n that never leave the non-negative orthant satisfies s n = S ( ) n · n − ( d − · "(cid:18) − A ( ) B ( ) (cid:19) (cid:18) S ( ) π (cid:19) d − p b · · · b d − O ( n − ) (cid:1) . (1)5ote that the result is trivial to apply to any given model, and is general enough to handle families ofmodels in varying dimension. Example 3.
Consider the step set where B (cid:0) z ˆ d (cid:1) = Q j Dominant asymptotics in the negative drift case are given by adding the asymptotic contributions of a finitecollection of points. Let ρ = q A ( ) B ( ) , and for each ≤ k ≤ d − define b k ( z ˆ k ) := [ z k ] S ( z ) = [ z − k ] S ( z ) . Furthermore, define C ρ := S ( , ρ ) ρ π d/ A ( )(1 − /ρ ) · s S ( , ρ ) d ρ b ( , ρ ) · · · b d − ( , ρ ) · B ( ) and let C − ρ be the constant obtained by replacing ρ by − ρ in C ρ (the term in the square-root will alwaysbe real and positive, so there is no ambiguity). Theorem 4 (Negative Drift Asymptotics) . Let S be a negative drift (weighted) step set that is symmetricover all but one axis and takes a step forwards and backwards in each coordinate. If Q ( z ˆ d ) = 0 (i.e., if thereare steps in S having z d coordinate ) then the number of walks of length n that never leave the non-negativeorthant satisfies s n = S ( , ρ ) n · n − d/ − · C ρ (cid:0) O ( n − ) (cid:1) . If Q ( z ˆ d ) = 0 then the number of walks of length n that never leave the non-negative orthant satisfies s n = n − d/ − · h S ( , ρ ) n · C ρ + S ( , − ρ ) n · C − ρ i (cid:0) O ( n − ) (cid:1) . Again, this result can be immediately applied to families of models. Note that S ( , ρ ) = Q ( ) +2 p A ( ) B ( ) . Example 5. Consider the step set where A (cid:0) z ˆ d (cid:1) = Q j Lattice Path Generating Functions We now show how to derive several useful expressions for the generating functions of the lattice path modelswe consider. We closely follow the kernel method as outlined in Bousquet-Mélou and Mishna [13], and astraightforward generalization to higher dimensions discussed by Melczer and Mishna [38]. This approachbuilds heavily on a probabilistic framework detailed by Fayolle, Iasnogorodski, and Malyshev [21]; see alsoBousquet-Mélou and Petkovšek [15] for a more general overview on the kernel method. To apply the kernel method we introduce the symmetry group of the walk. Definition 6. For ≤ j ≤ d − define the map σ j : C d → C d by σ j ( z , . . . , z d ) = ( z , . . . , z j − , z j , z j +1 , . . . , z d ) , and the map γ : C d → C d by γ ( z , . . . , z d ) = z , . . . , z d − , z d A (cid:0) z ˆ d (cid:1) B (cid:0) z ˆ d (cid:1) ! . We can view these maps as acting on Laurent polynomials f ∈ C [ z , z , . . . , z d , z d ] through σ · f ( z ) := f ( σ ( z , . . . , z d )) and further view them as acting on elements P n ≥ f n ( z ) t n ∈ C [ z , z , . . . , z d , z d ][[ t ]] by σ · X n ≥ f n ( z ) t n := X n ≥ ( σ · f n ( z )) t n = X n ≥ f n ( σ ( z )) t n . Finally, we let G be the group of birational transformations generated by σ , . . . , σ d − and γ . Remark 7. Since S is symmetric over all but one axis we have, for each j = 1 , . . . , d − , σ j (cid:0) A (cid:0) z ˆ d (cid:1)(cid:1) = A (cid:0) z ˆ d (cid:1) σ j (cid:0) B (cid:0) z ˆ d (cid:1)(cid:1) = B (cid:0) z ˆ d (cid:1) which, together with the fact that γ fixes S ( z ) , implies that S ( z ) is fixed by G . Furthermore, these equalitiesshow that the generators of G , which are involutions, commute, meaning G is the finite group of order d defined by G := n σ j · · · σ j d − d − γ j d : j , . . . , j d ∈ { , } o . The group G is the direct sum of d cyclic groups of order 2.Let F ( z , t ) be the multivariate generating function F ( z , t ) = X i ∈ N d n ≥ a i ,n z i t n , 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 endingat i ∈ N d recursively as a walk of length n − followed by a single step, one can show (see Melczer andMishna [38]) that the generating function satisfies a functional equation of the form (1 − tS ( z )) z F ( z , t ) = z + d X k =1 L k ( z ˆ k , t ) , L k ( z ˆ k , t ) ∈ Q [ z ˆ k ][[ t ]] . (2)7n particular, note that each L k ( z ˆ k , t ) is independent of the variable z k .When manipulating the formal expressions that arise in our application of the kernel method, we mayencounter rational functions in the variables z , . . . , z d which, in addition to not being analytic at the origin,are not Laurent polynomials in these variables. Thus, we make use of the iterated Laurent series ring R = Q (( z )) · · · (( z d ))[[ t ]] ; unless otherwise stated all computations below are assumed to take place in thering R , which contains both Q [ z , z , . . . , z d , z d ][[ t ]] and Q [[ z , t ]] . Note that every rational function in Q ( z ) has an expansion in R . For further details on iterated Laurent series, including their uses in combinatoricsand a classification of which formal series are iterated Laurent series, the reader is referred to the PhD thesisof Xin [52]. We define the positive sub-series extraction operator [ z ≥ ] : R → Q [[ z , t ]] by [ z ≥ ] X n ≥ X i ∈ Z d a i ,n z i t n := X n ≥ X i ∈ N d a i ,n z i t n . This setup leads to Theorem 8, typical of the kernel method (see Bousquet-Mélou and Mishna [13], forinstance, or Zeilberger [53] and Gessel and Zeilberger [29] for similar expressions in a multivariate setting). Theorem 8. If S is symmetric over all but one axis, then the multivariate generating function F ( z , t ) tracking endpoint and length satisfies F ( z , t ) = [ z ≥ ] P σ ∈G sgn( σ ) σ ( z . . . z d )( z · · · z d )(1 − tS ( z )) , (3) where sgn (cid:16) σ j · · · σ j d − d − γ j d (cid:17) = ( − j + ··· + j d . The generating function F ( z , t ) , and thus the specialized generating function F ( , t ) that counts the walks ofa given length ending anywhere, are D-finite. Note: The order of the iterated Laurent fields that define R is important. If one works in an iteratedLaurent field where z d is not the last variable before t , Equation (3) may not hold. Proof. We begin by examining the expression σ ( z . . . z d ) F ( σ ( z ) , t ) for some fixed σ = σ j · · · σ j d − d − γ j d ∈ G .When j d = 1 , then every term in the expansion of σ ( z . . . z d ) F ( σ ( z ) , t ) in the ring R will have negativepower of z d (due to the order of the variables used when defining R ). Otherwise, if j d = 0 and there is some k ∈ { , . . . , d − } such that j k = 1 then every term in the expansion of σ ( z . . . z d ) F ( σ ( z ) , t ) in the ring R will have negative power of z k . Thus, we see [ z ≥ ] σ ( z . . . z d ) F ( σ ( z ) , t ) = 0 for σ ∈ G unless σ is the identityelement. This implies [ z ≥ ] X σ ∈G sgn( σ ) σ ( z . . . z d ) F ( σ ( z ) , t ) = X σ ∈G sgn( σ ) (cid:0) [ z ≥ ] σ ( z . . . z d ) F ( σ ( z ) , t ) (cid:1) = ( z · · · z d ) F ( z , t ) . By definition, for all σ ∈ G and τ ∈ { σ , . . . , σ d − , γ } , sgn( τ σ ) = − sgn( σ ) . As S ( z ) is fixed by the elements of G , to prove Equation (3) from Equation (2) it is sufficient to show thatfor each k = 1 , . . . , d , X σ ∈G sgn( σ ) σ ( z . . . z d ) (cid:0) σ · L k ( z ˆ k , t ) (cid:1) = 0 . Fix k and write G as the disjoint union G = G ∪ G , where G = n σ j · · · σ j d − d − γ j d : j , . . . , j d ∈ { , } , j k = 0 o G = n σ j · · · σ j d − d − γ j d : j , . . . , j d ∈ { , } , j k = 1 o . g ∈ G , ( σ k g ) · L k ( z ˆ k , t ) = g · L k ( z ˆ k , t ) , and therefore X σ ∈G sgn( σ ) σ ( z . . . z d ) ( σ · L k ( z ˆ k , t )) = X σ ∈G sgn( σ ) σ ( z . . . z d ) ( σ · L k ( z ˆ k , t ))+ X σ ∈G sgn( σ ) σ ( z . . . z d ) ( σ · L k ( z ˆ k , t ))= X σ ∈G (sgn( σ ) − sgn( σ )) σ ( z . . . z d ) ( σ · L k ( z ˆ k , t ))= 0 , as desired. The results on D-finiteness follow from a classical result of Lipschitz [34] which states (in anequivalent form) that the class of D-finite functions is closed under positive sub-series extraction.The next result determines an explicit expression for the generating function under consideration. Lemma 9. For the group G , X σ ∈G sgn( σ ) σ ( z . . . z d ) = ( z − z ) · · · ( z d − − z d − ) z d − z d A (cid:0) z ˆ d (cid:1) B (cid:0) z ˆ d (cid:1) ! . Consequently, F ( z , t ) = [ z ≥ ] R ( z , t ) where R ( z , t ) = (1 − z − ) · · · (1 − z − d − ) (cid:0) B (cid:0) z ˆ d (cid:1) − z − d A (cid:0) z ˆ d (cid:1)(cid:1) B (cid:0) z ˆ d (cid:1) (1 − tS ( z )) . (4) Proof. The first statement follows directly from the definition of G and the sign operator (formally it can beproved by induction). The second statement comes from combining Lemma 9 with (3). Next, we turn back to the sequence counting the total number of walks of a given length (regardless ofendpoint). The generating function of this sequence is simply F ( , t ) , since specializing each z j variable to1 sums over its possible values.We may translate the positive sub-series extraction given by Equation (3) into an expression for F ( , t ) using the diagonal operator ∆ : R → Q [[ t ]] defined by ∆ X n ≥ X i ∈ Z d a i ,n z i t n := X n ≥ a n,...,n t n . Our asymptotic results will follow from an analysis of a diagonal expression for F ( , t ) . Establishing thisdiagonal expression is more complicated than in Melczer and Mishna [38], because we must consider expres-sions whose coefficients in t are not Laurent polynomials. In the completely symmetric case A = B in (4),and cancellation leaves only − tS ( z ) in the denominator.The following technical lemma is elementary, involving only algebraic manipulations (see also Melczerand Mishna [38, Proposition 2.6]). Lemma 10. Let P ( z , t ) ∈ Q [ z , z , . . . , z d , z d ][[ t ]] ⊂ R . Then (cid:0) [ z ≥ ] P ( z , t ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) z =1 ,...,z d =1 = ∆ (cid:18) P ( z , . . . , z d , z · · · z d · t )(1 − z ) · · · (1 − z d ) (cid:19) , (5) where the diagonal on the right hand side is taken as an expansion in R , as usual. P on the right hand side.We would like to use Lemma 10 directly, but R in (4) does not lie in the correct ring and the substitutionsindicated (replacing z i by z i ) are not formally justified. This problem can be circumvented through a te-dious but elementary generating function argument, taking into account the precise structure of the rationalfunction under consideration, yielding Proposition 11. Due to certain undesirable properties of the represen-tation (6) we use a different diagonal expression for the generating function more suited to an asymptoticanalysis, so we omit the proof. Proposition 11. Let S be a weighted step set satisfying the conditions above. Then the generating functioncounting the number of walks of a given length in the lattice path model defined by S satisfies F ( , t ) = ∆ (1 + z ) · · · (1 + z d − ) (cid:0) B (cid:0) z ˆ d (cid:1) − z d A (cid:0) z ˆ d (cid:1)(cid:1) (1 − z d ) B (cid:0) z ˆ d (cid:1) (1 − tz · · · z d S ( z , . . . , z d − , z d )) ! . (6)The rational function in (6) presents a challenge for the integral manipulations necessary to computeasymptotics as one can only easily deform domains of integration where the integrand is analytic; the factor B (cid:0) z ˆ d (cid:1) present in the denominator can give strange surfaces of singularities. Instead we use the followingalternative expression, which is a power series in t with Laurent polynomial coefficients in the other variables. Theorem 12. Let S be a weighted step set satisfying the conditions above. Then the generating functioncounting the number of walks of a given length in the lattice path model defined by S satisfies F ( , t ) = ∆ (cid:18) G ( z , t ) H ( z , t ) (cid:19) , where G ( z , t ) = (1 + z ) · · · (1 + z d − ) (cid:0) − tz · · · z d (cid:0) Q (cid:0) z ˆ d (cid:1) + 2 z d A (cid:0) z ˆ d (cid:1)(cid:1)(cid:1) H ( z , t ) = (1 − z d ) (cid:16) − tz · · · z d S ( z ) (cid:17)(cid:16) − tz · · · z d (cid:0) Q (cid:0) z ˆ d (cid:1) + z d A (cid:0) z ˆ d (cid:1)(cid:1) (cid:17) , (7) and S ( z ) = S ( z ˆ d , z d ) = z d B (cid:0) z ˆ d (cid:1) + Q (cid:0) z ˆ d (cid:1) + z d A (cid:0) z ˆ d (cid:1) . Proof. Expanding the expression in (4) we obtain (1 − z ) · · · (1 − z d − ) · − z d A (cid:0) z ˆ d (cid:1) B (cid:0) z ˆ d (cid:1) ! · X n ≥ t n (cid:0) z d A (cid:0) z ˆ d (cid:1) + Q (cid:0) z ˆ d (cid:1) + z d B (cid:0) z ˆ d (cid:1)(cid:1) n . As the sub-expression (1 − z ) · · · (1 − z d − ) · − z d A (cid:0) z ˆ d (cid:1) B (cid:0) z ˆ d (cid:1) ! · X n ≥ t n (cid:0) z d A (cid:0) z ˆ d (cid:1) + Q (cid:0) z ˆ d (cid:1)(cid:1) n contains no positive powers of z d , we can subtract it from R ( z ) and extract the positive part of (1 − z ) · · · (1 − z d − ) (cid:0) − z d A ( z ˆ d ) /B ( z ˆ d ) (cid:1) − tS ( z ) + (1 − z ) · · · (1 − z d − ) (cid:0) z d A ( z ˆ d ) /B ( z ˆ d ) (cid:1) − t ( z d A ( z ˆ d ) + Q ( z ˆ d )) . This final rational function simplifies to (1 − z ) · · · (1 − z d − ) (cid:0) − t (cid:0) z d A (cid:0) z ˆ d (cid:1) + Q (cid:0) z ˆ d (cid:1)(cid:1)(cid:1)(cid:0) − t (cid:0) z d A (cid:0) z ˆ d (cid:1) + Q (cid:0) z ˆ d (cid:1) + z d B (cid:0) z ˆ d (cid:1)(cid:1)(cid:1) (cid:0) − t (cid:0) z d A (cid:0) z ˆ d (cid:1) + Q (cid:0) z ˆ d (cid:1)(cid:1)(cid:1) , and we can now apply Lemma 10. 10ote that the power series expansion of /H ( z , t ) has all non-negative coefficients, which will allow us tosimplify 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 onour series coefficients we have lost some symmetry and less cancellation occurs. For example, the generatingfunction of the model with all possible steps in 2 dimensions is the diagonal of (1 + x )(1 + y )1 − t (1 + x + y + x + y + x y + xy + x y ) using the expression in Proposition 11, which coincides with that in [38], but the diagonal of (1 + x )(1 − t ( y + y + x y + xy + x y ))(1 − y )(1 − t (1 + x + y + x + y + x y + xy + x y ))(1 − t ( y + y + x y + xy + x y )) using the expression in Theorem 12. We end this section with a justification of why we only consider models missing one symmetry (instead oftwo, three, etc.). Indeed, as the following theorem shows, in any dimension d ≥ there exists a modelthat is missing two symmetries and admits a generating function that is not D-finite. As the diagonal ofa multivariate rational function must be D-finite [16, 34], this shows that it is impossible to determine theasymptotics of all models missing two symmetries uniformly through multivariate rational diagonals andanalytic combinatorics in several variables. Theorem 13. There exists a sequence of step sets S , S , . . . with S d defining a step set of dimension d thatis symmetric over all but two axes, such that the generating function of each model is non D-finite.Proof. If a counting sequence ( c n ) n ≥ has asymptotics of the form c n ∼ K · ρ n · n α for constants K, ρ, α ∈ R and its generating function is D -finite, then ρ is algebraic and α is rational (see Theorem 3 of Bostan,Raschel, and Salvy [10]).When d = 2 , consider the set of steps S = { ( − , − , (0 , − , (0 , , (1 , , ( − , } . Bostan, Raschel, and Salvy [10] show that ( e n ) n ≥ , the number of walks on these steps staying in the firstquadrant that begin and end at the origin, has dominant asymptotics e n ∼ K e · ρ ne · n α e where α is an irrational number (approximately 2.757466) equal to − − π/ arccos( − c ) , with c an algebraicnumber satisfying c − c + (3 / c − (1 / 8) = 0 . Work of Duraj [20] implies in our context that—since S has negative vector sum in both coordinates—the sequence counting the total number of steps has dominantasymptotics s (2) n ∼ K · ρ n · n α , where α = α e and is thus irrational.For d ≥ let S d = S × {± } d − . Every walk of length n on the steps S d is constructed uniquely from awalk of length n on the steps S in the non-negative quadrant and d − independent walks of length n on thesteps {− , } restricted to the non-negative integers (this is a simple version of the Hadamard decompositionof walks studied in Bostan et al. [4]). Thus, the number of walks of length n taking steps in S d restricted tothe d -dimensional non-negative orthant is s ( d ) n = s (2) n · c d − n , That article takes a probabilistic view of exit times for random walks to leave certain cones, and applies to a wide range ofmodels; see its Example 7 for the case of two dimensional random walks in a quadrant. c n is the number of Dyck paths that do not have to end at 0 (sometimes called Dyck prefixes). It isa classical result of enumerative combinatorics that c n = (cid:0) n ⌈ n/ ⌉ (cid:1) with dominant asymptotics of the form c n ∼ p /π · n · n − / , which implies s ( d ) n ∼ K d · ( ρ d ) n · n α d , with α d = α − d/ Q .It would be of great interest to find ‘simple’ diagonal expressions involving more general multivariatemeromorphic functions for walk models with non-D-finite generating functions. Such multivariate functionscould not be D-finite, in the sense that the vector space of all partial derivatives over Q ( z ) would need to beinfinite dimensional.Furthermore, although not all models missing two symmetries can be handled directly by our methods,there are some models missing two (or more) symmetries whose generating functions can be written asrational diagonals. For a specific model, one can attempt to follow the algebraic kernel method for higherdimensional walks; see, for instance, Bostan et al. [5] for a general framework. Characterizing all such modelsis a more difficult task. Our best guess is that this property is related to being a Hadamard decomposition,in the sense of Bostan et al. [4], of some (hopefully nice) characterizable family of ‘atomic’ D-finite models.Although in general one cannot simply determine asymptotics of a model which admits a Hadamard decom-position by multiplying the asymptotics of lower dimensional sequences, many properties such as D-finitenessare inherited via Hadamard decomposition; see Bostan et al. [4, Section 5] for details. Given a model whosegenerating function can be written as a rational diagonal, Courtiel et al. [17] develop a method to determineweightings of the step set so that the weighted generating function can be represented as a parametrizedrational diagonal with the weights as parameters.We now move on to the analysis of the expressions obtained using the methods of analytic combinatorics inseveral variables. We use the methods developed by Pemantle and Wilson [47] for asymptotics controlled bypoints where the zero set of H ( z , t ) is locally a manifold or a union of finitely many transversely intersectingmanifolds. Suppose Q ( z , t ) is a rational function analytic at the origin. As in the univariate case, a multivariatesingularity analysis starts from the Cauchy integral formula, which implies b n := [( z · · · z d t ) n ] Q ( z , t ) = 1(2 πi ) d +1 Z C Q ( z ) · d z dt ( z · · · z d t ) n +1 (8)for any n ∈ N and C a product of circles sufficiently close to the origin. If D is the domain of convergence ofthe power series of Q ( z , t ) at the origin, and V is the set of singularities of Q ( z , t ) , then any singularity onthe boundary ∂ D of D is called minimal . When P ( z , t ) is a polynomial we say that ( w , s ) is a minimal zero of P if P ( w , s ) = 0 and P ( y , r ) = 0 whenever | w j | ≤ | y j | , j = 1 , . . . , d, | r | ≤ | s | and one of the inequalities is strict. Note that a minimal point of Q is a minimal zero of its denominator,and vice-versa.As Q ( z , t ) is rational, b n grows at most exponentially and standard integral bounds imply lim sup n →∞ | b n | /n ≤ | w · · · w d s | − (9)12or any minimal point ( w , s ) ∈ ∂ D ∩ V . In the simplest cases, one hopes to identify a finite set of minimalpoints achieving the optimal bound in Equation (9). When such a set exists, and the local geometry of thealgebraic set V is sufficiently nice, asymptotics can then be determined.We now specialize our arguments to the rational function Q ( z , t ) = G ( z , t ) /H ( z , t ) defined by Theorem 12;note that G and H are co-prime, so the singularities of Q are the zeros V = V ( H ) of the polynomial H .Because of the nice form of H , we are able to characterize its minimal zeros achieving the best bound inEquation (9), which is typically the hardest step of any asymptotic analysis. We make use of the followingresult. Lemma 14. Suppose P ⊂ Z d is a finite set not contained in a hyperplane of R d , and a i > are positiveconstants for each i ∈ P . Then every critical point of P ( z ) = X i ∈P a i z i on ( R > ) d is a global minimum and P admits at most one critical point on this domain. Furthermore, sucha global minimum exists if and only if P is not contained in a halfspace containing the origin.Proof. This result follows from the strict convexity of the Laplace transform P ( e x , . . . , e x d ) ; for details seeGarbit and Raschel [26, Lemma 7].In order to reason about minimal zeros of H ( z , t ) , we define the factors H := 1 − tz · · · z d S ( z ) = 1 − tz · · · z d − (cid:0) z d A ( z ˆ d ) + z d Q ( z ˆ d ) + B ( z ˆ d ) (cid:1) H := 1 − tz · · · z d (cid:0) Q ( z ˆ d ) + z d A ( z ˆ d ) (cid:1) H := 1 − z d , and set V j = V ( H j ) .Under our assumptions on S the conditions of Lemma 14 are satisfied by S ( z ) , giving the following. Proposition 15. The unique minimal zero of H ( z , t ) with positive coordinates that minimizes | z · · · z d t | − is p := , , . . . , , s B ( ) A ( ) , p A ( ) /B ( )2 p A ( ) B ( ) + Q ( ) ! if the drift is negative p := (cid:18) , , . . . , , S ( ) (cid:19) otherwise . Proof. As | z · · · z d t | − decreases as ( z , t ) moves away from the origin, any such minimizer must be a zeroof H or H . Since S ( z ) has non-negative coefficients, any zero of H with positive coordinates is a minimalzero as t = ( z · · · z d S ( z )) − increases as one of the z j decreases and the others are constant. Furthermore,on V ∩ ( R > ) d | z · · · z d t | − = S ( z ) , which by Lemma 14 has a unique minimum corresponding to a unique critical point. The system S z ( z ) = · · · = S z d ( z ) = 0 can be reduced to (1 − z ) · [ z − ] S ( z ) = · · · = (1 − z d − ) · [ z − d − ] S ( z ) = B ( z ˆ d ) − z d A ( z ˆ d ) = 0 , which has only the solution ( , p B ( ) /A ( )) with positive coordinates, as S has all non-negative coefficients.13f the drift is non-positive, B ( ) ≤ A ( ) and p is a minimal zero of the product H ( z , t ) H ( z , t ) .Otherwise, any minimal zero of H ( z , t ) H ( z , t ) that minimizes | z · · · z d t | − must lie on V ∩ V , where | z · · · z d t | − = S ( z ˆ d , , and Lemma 14 implies the minimizer is p .Finally, if ( z , t ) ∈ V ∩ ( R > ) d then t = 1 z · · · z d (cid:0) Q ( z ˆ d ) + z d A ( z ˆ d ) (cid:1) > z · · · z d S ( z , t ) since z · · · z d B ( z ) > . But this implies ( z , t ) is not a minimal zero of H , as there exists ( z , s ) ∈ V with < s < t .In order to perform a local singularity analysis we will need to describe V near points of interest. In ourcase, the singular set V is the union of smooth manifolds V , V , and V (for each i , the gradient of H i nevervanishes when H i = 0 ). Furthermore, we show in the proof of Theorem 16 that any minimal singularity willnot lie on V , so that any minimal singularity is either in V alone, V alone, or the intersection V ∩ V . Asthe gradients of H and H are linearly independent at any common zero, we say V and V are transverse .In this setting, the stratum of minimal w ∈ V is the intersection of the V j containing w . Minimal w ∈ V with non-zero coordinates is called a minimal critical point if it is a critical point (in the differentialgeometry sense) of the map φ ( z ) = log( z · · · z d ) from the stratum of w to C . Algebraically, this means thatthe gradient of φ ( z ) = log( z · · · z d ) at z = w can be written as a linear combination of the gradients of the H j polynomials defining the strata of w . Critical points are those where the Cauchy integral can be locallymanipulated into a so-called Fourier-Laplace integral, where saddle-point methods can be applied to obtainasymptotics.General definitions of critical and contributing points, where local function behaviour dictates coefficientasymptotics, can be found in [47]. In particular, Proposition 10.3.6 of [47] gives an explicit characterizationof contributing points: in our setting, a singularity of Q ( z , t ) is a contributing point if it is a minimal criticalpoint that minimizes | z · · · z d t | − on ∂ D (the exponential order of the asymptotic contribution of that pointis maximum). Theorem 16. When the drift is positive, there are at most d − contributing points. The point p is one,and the others are the points ( w , , t ) where w ∈ {± } d − , t = 1 w · · · w d − · S ( w , , and | t | = 1 S ( , . When the drift is negative, there are at most d +1 contributing points. The point p is one, and theothers are the points ( w , w d , t ) where w ∈ {± } d − , w d = ν s B ( w ) A ( w ) , t = 1 w d S ( w , w d ) , | w d | = p B ( ) p A ( ) , and | t | = p A ( ) p B ( ) S (cid:16) , p A ( ) /B ( ) (cid:17) , with ν a fourth root of unity (note that in order to satisfy the condition on | t | it is necessary that B ( w ) /A ( w ) > , so the square root can be taken unambiguously).When the drift is zero, there are at most d contributing points. The point p = p is one, and the othersare the points ( w , w d , t ) where ( w , w d ) ∈ {± } d , t = 1 w · · · w d S ( w , w d ) , and | t | = 1 S ( , . roof. As the power series expansion of /H ( z , t ) has non-negative coefficients, every minimal point has thesame coordinate-wise modulus as an element of V with positive coordinates (an element of ∂ D is the limitof a sequence in D that makes the power series of /H approach infinity, but as the power series coefficientsare non-negative the series only gets larger when each coordinate is replaced by its modulus).Thus, we search for points in V with the same coordinate-wise modulus as p and p . First, we notethat no point in V is minimal as its t variable will be smaller than required. On V , we seek points ( w , s ) such that | S ( w , s ) | = S ( p ) or | S ( w , s ) | = S ( p ) . Since z · · · z d S ( z , t ) is a polynomial with non-negative coefficients, the triangle inequality implies the onlyway this can happen is if every monomial of S ( z , t ) has the same argument when evaluated at w . Ourassumptions on S imply that w , . . . , w d − must be real (and thus ± ) so the points in the statement ofTheorem 16 are the only potential minimizers of | z · · · z d t | − , and are minimal points.Computing the gradient of H ( z ) = 1 − tz · · · z d S ( z ) shows that a point ( z , t ) with stratum V is criticalif and only if z satisfies S z ( z ) = · · · = S z d ( z ) = 0 , while a point with stratum V ∩ V is critical if and only if S z ( z ) = · · · = S z d − ( z ) = 0 , z d = 1 . These equations are satisfied by the stated points, therefore we have found the set of contributing points. The results of Pemantle and Wilson [47] apply broadly to compute asymptotics when contributing pointsare known. Now that the set of contributing points is characterized by Theorem 16 it is a straightforward(though computationally intensive) matter to compute asymptotics, which we do for each of the cases inTheorem 16. Some of the technical and laborious proofs that are not of theoretical interest are given inAppendix A.We make an exponential change of variables to convert complex contour integrals to integrals over R d .To this end we introduce some basic notation. Definition 17. For p ∈ C d , define E on [ − π, π ] d by E p ( θ ) = ( p e iθ , . . . , p d e iθ d ) For fixed p , every function f ( z ) of z yields a corresponding function ˜ f ( θ ) := f ◦ E p of θ under this changeof variable.For ≤ j ≤ d , we use the usual notation ∂ j for the partial differential operator ( ∂/∂θ j ) and define thefunctions B ( z ˆ1 ) , . . . , B d − ( z [ d − ) by stipulating that B k is the unique Laurent polynomial such that S ( z ) = ( z k + z k ) B k ( z ˆ k ) + Q k ( z ˆ k ) (10)for some Laurent polynomial Q k . For notational convenience we set B d ( z ˆ d ) := B ( z ˆ d ) . Remark 18. We note that for each j = k with j, k < d, each B j is symmetric in z k and z k , and does notinvolve any power of z j . Also A, B, Q (and both S and S ) are symmetric in z k and z k .We also need the following quantities. Definition 19. For each j < d , define Laurent polynomials A ′ j , B ′ j , A ′′ j , B ′′ j by A ( z ˆ d ) = ( z j + z j ) A ′ j ( z ˆ j ) + A ′′ j ( z ˆ j ) B ( z ˆ d ) = ( z j + z j ) B ′ j ( z ˆ j ) + B ′′ j ( z ˆ j ) . f ( z ) we define ∇ log f ( z ) := ( z ∂ f, . . . , z d ∂ d f ) . We now have all the necessary tools to compute asymptotics, beginning with the positive drift case. In the positive drift case, when A ( ) < B ( ) , Theorem 16 implies that we are dealing with contributingpoints on the stratum V ∩ V . The next result follows from Theorem 10.3.4 of Pemantle and Wilson [47],where e j is the j th standard basis vector (with a 1 in its j th entry and zeros elsewhere). Proposition 20. Let Γ be the square matrix whose first rows are ∇ log H ( p ) and ∇ log H ( p ) , and whoselast d − rows are p j e j for j = 1 , . . . , d − . Furthermore, define g ( θ ) := log (cid:18) p · · · p d ) e i ( θ + ··· + θ d ) S ( p e iθ , . . . , p d e iθ d ) (cid:19) . Then [ t n ]∆ Q ( z , t ) = p n · n − ( d − / · (2 π ) − ( d − / ( d + 1) − ( d − / G ( p )det Γ · p det g ′′ ( ) + O ( n − ) ! , where g ′′ ( ) denotes the Hessian of g ( θ ) at the origin.Proof. As p is the only contributing point where G ( z , t ) does not vanish, Theorem 10.3.4 of Pemantle andWilson [47] gives the above asymptotic result and the bound of O ( n − ) on the lower order terms.Because G ( z , t ) vanishes at all contributing points except for p , no positive drift model will have peri-odicity in its leading asymptotic term. Applying Proposition 20 in our situation gives asymptotics in thepositive drift case. Proof of Theorem 2. Theorem 16 implies that the point p = ( , , /S ( , is the unique contributingpoint at which G ( z , t ) does not vanish. At this point, one can calculate that Γ = · · · − − − − · · · − − r − 11 0 0 · · · · · · · · · ... . . . . . . . . . . . . ... ... · · · for a real number r < , which does not appear in the determinant of Γ , and G ( , / | S | ) H ( , / | S | ) = 2 d − ( B ( ) − A ( )) S ( ) S ( ) B ( ) = 2 d − (cid:18) − A ( ) B ( ) (cid:19) . The Chain Rule and Lemma 27 of Appendix A imply that g ′′ ( ) is the diagonal matrix g ′′ ( ) = B ( )( d +1) S ( ) · · · B ( )( d +1) S ( ) · · · ... . . . . . . . . . ... · · · B d − ( )( d +1) S ( ) 00 0 · · · B d − ( )( d +1) S ( ) , 16o that Proposition 20 gives s n = S ( ) n n − / (2 π ) − d − ( d + 1) − ( d − · d − (cid:16) − A ( ) B ( ) (cid:17)p ( d + 1) − ( d − d − S ( ) − ( d − / ( b · · · b d − ) + O ( S ( ) n n − / ) , which simplifies to (1). In the negative drift case, when A ( ) < B ( ) , Theorem 16 implies that we are dealing with contributingpoints on the stratum V where V itself is locally a manifold. This simplifies computations, but an addeddifficulty is that the numerator vanishes to at least first order at every critical point. We now state ageneral theorem that allows one to calculate asymptotics under these conditions, coming from Raichev andWilson [49] (note that for us the dimension d is one less than the number of variables of F ). Theorem 21. Fix natural number N > and recall the above notation from this section. In a neighborhoodin V of a smooth critical point p on V , write t = h ( z ) . Define u, ˜ g and ˜ g by u ( z ) = − G ( z , h ( z )) h ( z )( ∂H/∂t )( z , h ( z ))˜ g ( θ ) = log ˜ h ( θ )˜ h ( ) + i d X j =1 θ j ˜ g ( θ ) = ˜ g ( θ ) − θ T ˜ g ′′ ( ) θ . Supposing that the Hessian determinant det ˜ g ′′ ( ) = , define Ψ ( p ) n,N := p − n · n − d/ · (2 π ) − d/ (det ˜ g ′′ ( )) − / N − X k =0 n − k L k (˜ u, ˜ g ) , (11) where L k (˜ u, ˜ g ) = k X l =0 H k + l (˜ u ˜ g l )( )( − k k + l l !( k + l )! with H the differential operator H = − X ≤ a,b ≤ d (˜ g ′′ ( ) − ) a,b ∂ a ∂ b . Then, as n → ∞ , [ t n ]∆ Q ( z , t ) = X p ∈ W Ψ ( p ) n,N + O ( n − N ) . Lemma 28 of Appendix A shows that the term L (˜ u, ˜ g ) , which will determine dominant asymptotics fornegative drift models, simplifies considerably for the functions we consider. In the setting of this section wehave ˜ u ( θ ) = (1 + p e iθ ) · · · (1 + p d − e iθ d − ) (cid:0) − p d e iθ d A ( p ˆ d ) /B ( p ˆ d ) (cid:1) − p d e iθ d ˜ g ( θ ) = log S ( p ) − log S ( p e iθ , . . . , p d e iθ d ) , ˜ g ′′ ( ) is the diagonal matrix ˜ g ′′ ( ) = p B ( p ˆ1 ) S ( p ) · · · p B ( p ˆ2 ) S ( p ) · · · ... . . . . . . . . . ... · · · p d − B d − ( p [ d − ) S ( p ) 00 0 · · · B d ( p ˆ d ) p d S ( p ) . Extensive calculations, given by Proposition 29 and Proposition 30 in Appendix A, then give the following. Proposition 22. Let S be a step set that is symmetric over all but one axis and takes a step forwards andbackwards in each coordinate, and let W be the set of contributing points given by Theorem 16. If S hasnegative drift, then the number of walks of length n that never leave the non-negative orthant satisfies s n = X p ∈ W h ( p · · · p d p t ) − n n − d/ − (cid:0) K p C p + O ( n − ) (cid:1)i , (12) where K p = 2 − d π − d/ S ( p ) d/ (cid:16) p · · · p d − · B ( p ˆ1 ) · · · B d − ( p [ d − ) B ( p b d ) /p d (cid:17) − / C p = S ( p ) Q j 32 : n odd , B n = ( √ n even 18 : n odd , C n = ( √ 30 : n even / √ n odd Four of the models in Table 2 have step sets that are symmetric over every axis. This means that theirasymptotics follow directly from the work of Melczer and Mishna [38] (see Theorem 1 above). The asymptoticorder is |S| n n − in each case. There are six models whose step sets are missing one symmetry and have positive drift; one can directlyapply Theorem 2 to prove the asymptotics listed. The asymptotic order is |S| n n − / in each case. There are six models whose step sets are missing one symmetry and have negative drift; one can applyTheorem 4 to prove the asymptotics listed (we note that the original table of guessed asymptotics by Bostanand Kauers [7] has small errors in the constants for the first three of these models). The asymptotic orderis (cid:16) Q ( ) + 2 p A ( ) B ( ) (cid:17) n n − in each case. This was later proved using non-computer based arguments by Bostan, Kurkova, and Raschel [9] and Bousquet-Mélou [12]. xample 23. Consider the model defined by step set S = { (0 , , ( − , − , (0 , − , (1 , − } = { N, SE, S, SW } . Here we have G ( z , t ) = (1 + x ) (cid:0) − y ( x + 1 + x ) (cid:1) / (1 − y ) H ( z , t ) = 1 − t ( x + y + xy + x y ) , and two of the eight possible points described by Theorem 16 are contributing points: p = (1 , / √ , / and p = (1 , − / √ , / . Using Sage to implement Proposition 29, we can calculate the contribution at eachcontributing point to be Ψ ( p ) n = 3 √ √ π · (2 √ n n Ψ ( p ) n = 3 √ − √ π · ( − √ n n so that the number of walks of length n satisfies s n = (2 √ n n · √ π (cid:16) √ − ( − n ) + 2(1 + ( − n ) + O ( n − ) (cid:17) = ( (2 √ n n · (cid:16) √ π + O ( n − ) (cid:17) : n even (2 √ n n · (cid:0) π + O ( n − ) (cid:1) : n odd The first three models here each have two contributing points that determine the dominant asymptotics,giving a periodicity in the coefficients as seen in the above example. There are four models for which the orbit sum method fails to give an expression for the walk generatingfunctions as the diagonals of rational functions (meaning the techniques of analytic combinatorics in severalvariables as described above cannot by directly used). Luckily, these four models are algebraic and explicitminimal polynomials for the generating functions are known: the first was found by Mishna [41], the nexttwo by Bousquet-Mélou and Mishna [13] and the final model—known as Gessel’s walk—is treated in Bostanand Kauers [8]. It is effective to determine the asymptotics of a sequence from its generating function’sminimal polynomial (under gentle technical conditions), so the asymptotics for these cases follow rigorouslythrough univariate methods (see Chapter VII.7 of Flajolet and Sedgewick [25] and [41, 8]). In fact, themultivariate generating function enumerating walks by length and endpoint is algebraic for each model, astronger property. There are three models not covered by the above cases. They do not exhibit any symmetries, but theorbit sum method still gives a rational diagonal expression. Asymptotics of these models were previousgiven by Bousquet-Mélou and Mishna [13]: although their multivariate generating functions enumeratingwalks by length and endpoint are transcendental, the first two models have univariate generating functions F ( t ) counting walks ending anywhere in the quadrant which are algebraic. The final model does not havean algebraic univariate generating function, but Bousquet-Mélou and Mishna determined asymptotics byexploiting the fact that the coefficients of its multivariate generating function are Gosper summable ; see [13,Proposition 11] for details. 20 .5.1 Case 1: S = { N, W, SE } Applying the kernel method, the counting generating function satisfies F ( t ) = ∆ (cid:18) ( x − y )(1 − xy )( x − y )(1 − x )(1 − y )(1 − xyt ( y + yx + x )) (cid:19) . Furthermore, the kernel method implies (see, for example, Bousquet-Mélou and Mishna [13] or Bostan etal. [6]) that F ( t ) = 1 − t − √ − t − t t is algebraic, and asymptotics can be determined directly from this specification. Alternatively, one canperform a (more difficult) multivariate singularity analysis on this rational function. Although the rationalfunction has smooth and transverse multiple points that are minimal and critical, we cannot directly applythe asymptotic methods discussed above at the point (1 , , / —which turns out to be a contributingsingularity—as the gradient of H at that point is parallel to the gradient of the function φ ( x, y, t ) = log( xyt ) occurring in the Fourier-Laplace integral that must be analyzed to determine asymptotics. The theory inthis case, where three factors in the denominator intersect, still needs to be fully developed. Here, one canwrite x − y = ( x − x + 1) − ( y − to decompose the rational function as ( x − y )(1 − xy )( x − y )(1 − x )(1 − y )(1 − xyt ( y + yx + x )) = − (1 − xy )( x − y )( x + 1)(1 − y )(1 − xyt ( y + yx + x ))+ (1 − xy )( x − y )(1 − x )(1 − xyt ( y + yx + x )) , where each of the summands now contains only two factors in the denominator. After this simplification, a(still difficult) asymptotic analysis can be applied to determine the asymptotic contribution of each summandto the diagonal sequence. Using the explicit algebraic expression, or the multivariate approach, gives thatthe counting sequence for the number of walks on these steps satisfies s n = 3 n n / √ √ π + O (cid:0) n − (cid:1)! . It is interesting that the D-finite models which are harder to approach using multivariate analytic methodsare precisely those which have algebraic generating functions; we do not see a deep reason why this shouldbe true. S = { N W, SE, N, S, E, W } Applying the kernel method, we see that the counting generating function satisfies F ( t ) = ∆ (cid:18) ( x − y )(1 − xy )( x − y )(1 − x )(1 − y )(1 − txy ( x + y + xy + yx + x + y )) (cid:19) . Analogously to the last case, one can compute an algebraic expression F ( t ) = 1 − t − √ − t − t t for the generating function, which immediately gives asymptotics, or decompose the multivariate rationalfunction into a sum of rational functions with simpler singular sets and perform a multivariate analysis. Ineither case, one obtains that the counting sequence for the number of walks on these steps satisfies s n = 6 n n / √ √ π + O (cid:0) n − (cid:1)! . .5.3 Case 3: S = { E, SE, W, N W } Applying the kernel method, we see that the counting generating function satisfies F ( t ) = ∆ (cid:18) ( x + 1)( x − y )( x − y )( x + y )1 − xyt ( x + xy + yx + x ) (cid:19) . This case turns out to be easy to analyze, since the denominator is smooth. There are two points that satisfythe critical point equations: p = (1 , , / and p = ( − , , / , both of which are minimal and smooth.As the numerator has a zero of order 2 at p but order 3 at p , in fact only p contributes to the dominantasymptotics. The Sage package of Raichev—implementing Theorem 21—computes the contributions at bothpoints and shows that the counting sequence for the number of walks on these steps satisfies s n = 4 n n · π + O (cid:18) n n (cid:19) Weighted versions of this step set were studied in detail by Courtiel et al. [17]. We end with some additional remarks and generalizations. Although our results hold for models whose steps have positive real weights, we have not yet given an examplewith positive weights not equal to one. We do so now. Example 24. Consider the general 2D model with one symmetry defined by the following step set withnon-negative real weights, dab ec dab Then s n is asymptotic to (2( a + b + d ) + c + e ) n n − / · " (cid:18) − d + e b + c (cid:19) s(cid:18) a + b + d ) + c + e ( a + b + d ) π (cid:19) when b + c > d + e > (positive drift), whereas for < b + c < d + e (negative drift) and a = 0 it isasymptotic to (cid:16) a + 2 p (2 b + c )(2 d + e ) (cid:17) n n − · C with C = (2 a + 2 p (2 b + c )(2 d + e )) π (cid:16) − q b + c d + e (cid:17) ((2 b + c )(2 d + e )) / r d q b + c d + e + a + b q d + e b + c . Asymptotics in the subcase when < b + c < d + e (negative drift) and a = 0 similarly follows fromTheorem 4, resulting in an unwieldy formula due to periodicity. Finally, when b + c = 2 d + e (zero drift)and b = d and c = e (highly symmetric) then s n is asymptotic to (2 a + 2 c + 4 b ) n n − · " a + 2 c + 4 bπ p ( a + 2 b )( c + 2 b ) . he non-highly symmetric zero drift cases are outside the scope of our results. Conjecturally, they havedominant asymptotics which are a constant times (2 a + 2 b + 2 c + 2 d ) n n − for generic a, b, c, d , although thereare values of the parameters for which this does not hold. The techniques of analytic combinatorics in several variables are currently at the front line of research intocomputability questions in enumerative combinatorics. Given a univariate rational generating function, oran algebraic generating function encoded by its minimal polynomial and a sufficient number of initial terms,there are algorithms that take the function and return asymptotics of its power series coefficients at theorigin. On the other hand, it is an open problem whether it is decidable to take a D-finite generating functionencoded by an annihilating linear differential equation and initial conditions and determine asymptotics ofits counting sequence. In slightly restricted settings (for instance, when the D-finite generating function hasinteger coefficients and positive radius of convergence) a careful singularity analysis allows one to determinea so-called asymptotic basis : a finite collection of terms ∆ , . . . , ∆ d with asymptotic expansions of the form ∆ j = ρ nj n α j (log n ) κ j C ( j )0 + C ( j )1 n + · · · ! that can be determined explicitly to any finite order, such that asymptotics of the coefficient sequence c n isan R − linear combination of the ∆ j , c n ∼ K ∆ + · · · + K r ∆ r , K j ∈ R . See Flajolet and Sedgewick [25, Sec VII. 9] for details.In this way, decidability of asymptotics can be reduced to the determination of the connection coefficients K j . If, without loss of generality, asymptotics of ∆ dominates asymptotics of the other ∆ j then ∆ typicallydetermines (up to a scaling multiple) asymptotics of c n . However, if the constant K is zero then c n canhave drastically different asymptotic behaviour than ∆ . Determining the coefficients K j is known as the connection problem .Because the class of multivariate rational diagonals contains the class of algebraic functions, and iscontained in the class of D-finite functions, the techniques of analytic combinatorics in several variables offertools to investigate the connection problem (see Melczer [36] for an in-depth look at this approach). Forinstance, Bostan et al. [6] give annihilating differential equations for each lattice path generating functionin Table 2, even representing them in terms of explicit hypergeometric functions; however, they were notable to prove all asymptotics in that table, because of the connection problem. For instance, they show [6,Conjecture 2] that the number of walks with step set S = { (0 , − , ( − , , (1 , } has dominant asymptoticsof the form √ √ π k k − / if and only if the integral I := Z / (cid:26) (1 − v ) / v (1 + v ) / (cid:20) − v ) · F (cid:18) / , / (cid:12)(cid:12)(cid:12)(cid:12) v (cid:19) + 6 v (3 − v + 14 v ) · F (cid:18) / , / (cid:12)(cid:12)(cid:12)(cid:12) v (cid:19)(cid:21) − v + 4 v (cid:27) dv has the value I = 1 (see that paper for details on the notation used). Using the multivariate singularityanalysis discussed above, we are able to circumvent these difficulties, and resolve the connection problem forthese lattice path models. As an indirect corollary of our asymptotic results, we thus determine the valuesof certain complicated integral expressions involving hypergeometric functions.23 .3 Walks returning to boundaries The kernel method as presented here uses the multivariate generating function F ( z , t ) tracking walk lengthand endpoint to derive a rational diagonal expression for the univariate generating function F ( , t ) countingthe number of walks ending anywhere. Also of interest is the number of walks ending on one or more of theboundary hyperplanes in the first orthant; if V ⊂ { , . . . , d } then F ( z , t ) (cid:12)(cid:12)(cid:12) z j =0 ,j ∈ Vz j =1 ,j / ∈ V counts the number of walks returning to the intersection of the boundary hyperplanes { z j = 0 } for j ∈ V .Lemma 10 can easily be generalized to the following. Lemma 25. Let P ( z , t ) ∈ Q [ z , z , . . . , z d , z d ][[ t ]] ⊂ R . Then (cid:0) [ z ≥ ] P ( z , t ) (cid:1) (cid:12)(cid:12)(cid:12) z j =0 ,j ∈ Vz j =1 ,j / ∈ V = ∆ P ( z , . . . , z d , z · · · z d · t )(1 − z ) · · · (1 − z d ) · Y j ∈ V (1 − z j ) . Thus, following the arguments above, if Q ( z , t ) = G ( z , t ) /H ( z , t ) is the rational function given in Theo-rem 12 such that F ( , t ) = (∆ Q )( t ) then F ( z , t ) (cid:12)(cid:12)(cid:12) z j =0 ,j ∈ Vz j =1 ,j / ∈ V = ∆ Q ( z , t ) · Y j ∈ V (1 − z j ) . This close link between the diagonal expressions for walks ending anywhere and walks ending on boundaryhyperplanes allows us to reuse much of the work above to derive asymptotics for walks ending on boundaryhyperplanes. In particular, if V does not contain d then the singular sets of both multivariate rationalfunctions obtained are the same, so the contributing points calculated by Theorem 16 are still contributing.Analysis of asymptotics is easy for any fixed model, but the additional zeros in the numerator of Q ( z , t ) · Q j ∈ V (1 − z j ) at contributing points make explicit expressions for generic models harder to calculate.When V contains d , then the factor of − z d in the numerator of Q ( z , t ) will cancel the new factor − z d inthe numerator. In the negative drift and zero drift cases this has no bearing on the contributing singularities,and hence on the exponential growth of the number of walks returning to the hyperplane { z d = 0 } . Howeverin the positive drift case the contributing singularities will change and the exponential growth will be smallerfor walks returning to the hyperplane { z d = 0 } than for general walks.Using the Sage package of Raichev to compute asymptotic contributions, Table 3 gives asymptotics forthe number of walks returning to one or both of the boundary axes on the 2D quadrant models analyzedabove, where δ n = ( n ≡ otherwise σ n = ( n ≡ otherwise ǫ n = ( n ≡ otherwise and γ n = √ n ≡ n ≡ √ n ≡ n ≡ help account for periodicities that appear, and the algebraic constants A, B, and C are given by A = (156 + 41 √ q − √ , B = (583 + 138 √ q − √ , C = (4571 + 1856 √ q − √ . This completes, for the first time, a proof of conjectured asymptotics given by Bostan et al. [6]. Note thatthe second and third columns show that periodicity can occur even with positive drift models (unlike thesituation for walks ending anywhere analyzed in previous sections).24 Return to x -axis Return to y -axis Return to origin π · n n π · n n δ n π · n n δ n π · n n δ n π · n n δ n π · n n √ π · n n δ n √ π · n n δ n √ π · n n π · n n π · n n π · n n √ √ π n n / δ n √ π (2 √ n n ǫ n √ π (2 √ n n √ π n n / δ n √ π (2 √ n n δ n √ π (2 √ n n √ √ π n n / √ √ / π (2+2 √ n n √ / π (2+2 √ n n √ √ π n n / δ n √ π (2 √ n n δ n √ π (2 √ n n √ √ π n n / √ √ / π (2+2 √ n n √ / π (2+2 √ n n √ √ π n n / A π (2+2 √ n n B π (2+2 √ n n γ n · π · (2 √ n n δ n √ π (2 √ n n ǫ n √ π (2 √ n n (cid:16) δ n √ π + δ n − π (cid:17) · (2 √ n n δ n √ π (2 √ n n δ n √ π (2 √ n n √ / π · (2+2 √ n n √ √ / π (2+2 √ n n √ / π (2+2 √ n n (cid:16) δ n √ π + δ n − √ π (cid:17) · (2 √ n n δ n √ π (2 √ n n δ n √ π (2 √ n n √ / π · (2+2 √ n n √ √ / π (2+2 √ n n √ / π (2+2 √ n n C π · (2+2 √ n n A π (2+2 √ n n B π (2+2 √ n n √ √ π · n n / √ √ π · n n / σ n √ π · n n δ n π · n n π · n n δ n π · n n √ √ π · n n / √ √ π · n n / √ π · n n Table 3: Asymptotics of quadrant walks returning to the x -axis, the y -axis, and the origin, respectively. In the non-highly symmetric zero drift case, when A ( ) = B ( ) but A ( z ˆ d ) = B ( z ˆ d ) , Theorem 16 impliesthat we can have contributions from the point p := p = p on the stratum V ∩ V , possibly with otherpoints lying on locally smooth parts of V . Note that the numerator vanishes to at least first order at everycritical point, and that this case cannot occur for unweighted steps in dimension (where every zero driftmodel is highly symmetric).Since p is on the intersection of V and V , and the numerator vanishes, we expect it to give an asymptoticcontribution of C · |S| n · n − d/ − / , while the other (locally smooth) contributing points have a contributionof O (cid:0) |S| n · n − d/ − (cid:1) . Thus, if we can determine a second order contribution at p and show that it does notvanish, we will have found dominant asymptotics. 25symptotic contributions of minimal critical points are determined by analyzing integrals of the form Z [ − , r A ( θ ) e − nφ ( θ ) d θ where r ∈ N , and A and φ analytic functions from [ − , r to C (see [47] for details). When the gradient of φ vanishes in the interior of [ − , r , and other technical conditions on A and φ (which are satisfied here) hold,the asymptotic formulas in Theorem 21 follow. Unfortunately, in the non-highly symmetric zero drift casethe gradient of φ vanishes on the boundary of the domain of integration, meaning the relevant asymptoticconstants are not the same as those in Theorem 21. In fact, general asymptotics for such a situation havenot yet been worked out in the context of ACSV.Furthermore, while non-vanishing of the second order contribution at p happens generically, there aremodels where vanishing does occur and finding dominant asymptotics requires a detailed analysis at severalcontributing singularities. Because of these added difficulties, including a need to extend the underlyinganalytic theory, a more nuanced study of the zero drift models will be the subject of future work. As we have seen, the kernel method shows how nice combinatorial properties of a step set (like symmetryover axes) correspond to nice analytic properties of a multivariate rational function (like a singular set de-fined as the union of a small number of smooth manifolds) encoding the corresponding generating function.Furthermore, it is possible to turn this around: because diagonal sequences of multivariate rational func-tions with ‘simple’ geometry at contributing singularities can only capture a restricted set of asymptoticbehaviour, certain step sets whose asymptotics are sufficiently complicated cannot have their generatingfunctions encoded as the diagonals of ‘nice’ rational functions.The connection between analytic and combinatorial behaviour also helps explain patterns in asymptotics.For instance, it was previously observed that the exponential growth of 2D quadrant walks ending anywhereand the exponential growth of walks ending at the origin was the same for negative drift models but differentfor positive drift models. 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Asymptotics of coefficients of multivariate generating functions: Improve-ments for smooth points. Electr. J. Comb. , 15(1), 2008.[50] K. Raschel. Counting walks in a quadrant: a unified approach via boundary value problems. J. Eur.Math. Soc. (JEMS) , 14(3):749–777, 2012.[51] T. Tate and S. Zelditch. Lattice path combinatorics and asymptotics of multiplicities of weights intensor powers. J. Funct. Anal. , 217(2):402–447, 2004.[52] G. Xin. The Ring of Malcev-Neumann Series and the Residue Theorem . PhD thesis, Brandeis University,5 2004.[53] D. Zeilberger. André’s reflection proof generalized to the many-candidate ballot problem. DiscreteMath. , 44(3):325–326, 1983. A Calculus Computations General Results Here we collect the results and proofs that are necessary for determining asymptotics but theoreticallyuninteresting. Our first two results will be useful for calculating derivatives. Lemma 26. Let P, Q be smooth functions from I ⊂ R d to C , and suppose that lies in the interior of I and Q ( ) = 0 . Let ∂ := ∂ k be a partial derivative operator such that ( ∂P )( ) = 0 = ( ∂Q )( ) .Then ∂ ( P/Q )( ) = 0 and ∂ ( P/Q )( ) = Q ( ) ∂ P ( ) − P ( ) ∂ Q ( ) Q ( ) . When P ( θ ) = (cid:0) e iθ k + e − iθ k (cid:1) P k ( θ ˆ k ) + R k ( θ ˆ k ) Q ( θ ) = (cid:0) e iθ k + e − iθ k (cid:1) Q k ( θ ˆ k ) + R ′ k ( θ ˆ k ) then ( ∂P )( ) = 0 = ( ∂Q )( ) and ∂ ( P/Q )( ) = − P k ( ) Q ( ) + 2 P ( ) Q k ( ) Q ( ) . Proof. The first assertion follows from expanding the second derivative using the quotient rule, and applyingthe obvious simplification. The second follows from the first by direct substitution, since ( ∂ P )( ) simplifiesto − P k ( ) and similarly for Q . 29 emma 27. For a point p ∈ C d , let ˜ S ( θ ) := ˜ S ( θ ) = S ( p e iθ , . . . , p d e − iθ d ) . Then if p ˆ d ∈ {± } d − and p d = ± p B ( p ˆ d ) /A ( p ˆ d ) , we have for ≤ j ≤ k ≤ d : ∂ j ˜ S ( ) = 0 and ∂ j ∂ k ˜ S ( ) = if j = k ; − B d ( p ˆ d ) /p d : if j = k = d ; − p j B j ( p ˆ j ) : if j = k < d. Furthermore, if j < d then ∂ d ∂ j ˜ S ( ) = − ip j (cid:16) p d ˜ A ′ j ( ) − p − d ˜ B ′ j ( ) (cid:17) . If p ′ d = i p d is the imaginary number corresponding to p d , and S ( , 1) = S ( p ˆ d , p ′ d ) , then the values of allpartial derivatives above equal the values of the derivatives calculated for ( p ˆ d , p ′ d ) , potentially up to sign.Proof. For j < d , applying ∂ j to ˜ S ( θ ) = ( p j e iθ j + p − j e − iθ j ) ˜ B j ( θ ) + ˜ Q j ( θ ) yields ∂ j ˜ S ( θ ) = ( ip j e iθ j − ip − j e − iθ j ) ˜ B j ( θ ) . Note that from this point, applying any ∂ k with k = j will not change the factor ip j e iθ j − ip − j e − iθ j . Also,evaluating at zero gives i ( p j − p − j ) = 0 .Repeating this with higher powers of ∂ j gives a formula that is periodic in the exponent, with period .In particular when we evaluate at we obtain ∂ nj ˜ S ( ) = ( ( − n/ p j ˜ B j ( ) if n is even if n is odd . A similar computation with j = d yields ∂ nd ˜ S ( ) = ( ( − n/ B ( ) /p d if n is even if n is odd . Finally, consider the third order derivative ∂ d ∂ j ˜ S . Writing ˜ S ( θ ) = p d e iθ d A ( θ ) + Q + p − d e − iθ d B ( θ ) and differentiating using the formulae in Definition 19 yields the stated result.The statement about p ′ d follows from the same considerations, using the fact that if S ( , 1) = S ( p ˆ d , p ′ d ) then B ( p ˆ d , p ′ d ) /p ′ d and A ( p ˆ d , p ′ d ) p ′ d have the same argument. Negative Drift Model Calculations We now show that the quantities appearing in Theorem 21 simplify for us. Since in our situation of interestwe always have L (˜ u, ˜ g ) = ˜ u ( ) = 0 , we begin by considering the term corresponding to k = 1 in (11). Forpossible independent interest we show that some simplification is possible even in the general case.30 emma 28. In the general smooth case L (˜ u, ˜ g ) = − H (˜ u )( ) + H (˜ u ˜ g )( )4 + ˜ u H (˜ g )( )48 ! . If ˜ u vanishes to order at least at , then L (˜ u, ˜ g ) = − H (˜ u )( ) + H (˜ u ˜ g )( )4 ! and only terms involving third partial derivatives of ˜ g contribute to the H term.If ˜ u vanishes to order at least at then L (˜ u, ˜ g ) = − H (˜ u )( ) . If ˜ u vanishes to order at least at then L (˜ u, ˜ g ) = 0 . Proof. First note that any partial derivative of order at most of ˜ g is zero when evaluated at , by con-struction. Furthermore all derivatives of ˜ g of degree more than yield the same result when evaluated at as the corresponding derivative of ˜ g , since the difference between the two functions is quadratic. Thus ineach nonzero term in an expansion of L we may replace ˜ g by ˜ g .Since ˜ g vanishes to order at , the term involving H simplifies substantially, since in order to obtaina nonzero term all the th partial derivatives must be applied to ˜ g and so H (˜ u ˜ g ) simplifies to ˜ u H (˜ g ) .Similarly, H (˜ u ˜ g ) simplifies, since each 4th order partial derivative, when applied to the product ˜ u ˜ g andthen evaluated at , only yields a nonzero result when at least of the derivations are applied to ˜ g . If ˜ u vanishes to order then even these terms are zero. If ˜ u vanishes to order then it is exactly the 3rd partialsof ˜ g that can contribute.This, combined with Theorem 21, directly gives the following. Proposition 29. Let S be a step set that is symmetric over all but one axis and takes a step forwards andbackwards in each coordinate, and let W be the set of contributing points determined by Theorem 16. If S has negative drift, then the number of walks of length n that never leave the non-negative orthant satisfies s n = X p ∈ W Ψ ( p ) n (13) for Ψ ( p ) n = ( p · · · p d p t ) − n h n − d/ − K p C p + O ( n − ) i , where K p = 2 − d π − d/ S ( p ) d/ (cid:16) p B ( p ˆ1 ) · · · p d − B d − ( p [ d − ) B d ( p ˆ d ) /p d (cid:17) − / ,C p = − H (˜ u )( ) + H (˜ u ˜ g )( )4 ! for differential operator H = − S ( p )2 p d B d ( p ˆ d ) ∂ d + X j In the situation of Proposition 29, we have C p = S ( p ) Q j Note that ∂ k ˜ g = − ∂ k ˜ S/ ˜ S . This evaluates to zero at . It follows from Lemma 26 that ∂ nk ˜ g evaluatesat to − ∂ nk ˜ S ( ) / ˜ S ( ) . Also, when we evaluate ∂ d ∂ j ˜ g at , it simplifies to ∂ d ∂ j ˜ S ( ) / ˜ S ( ) .Now define X := Y j We first seek to compute − H (˜ u )( ) = S ( p )4 p d B d ( p ˆ d ) ∂ d ˜ u ( ) + X j