Lattice path enumeration using diagonals

Asymptotic lattice path enumeration using diagonals

Stephen Melczer  and  Marni Mishna Cheriton School of Computer Science, University of Waterloo, Waterloo ON Canada & U. Lyon, CNRS, ENS de Lyon, Inria, UCBL, Laboratoire LIP smelczer@uwaterloo.ca Department of Mathematics, Simon Fraser University, Burnaby BC, Canada, V5A 1S6 mmishna@sfu.ca
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.

Key words and phrases:
Lattice path enumeration, D-finite, diagonal, analytic combinatorics in several variables, Weyl chambers

1. Introduction

The reflection principle and its various incarnations have been indispensable in the study of the lattice path models, particularly in the discovery of explicit enumerative formulas. Two examples include the formulas for the family of reflectable walks in Weyl chambers of Gessel and Zeilberger [GeZe92], and various approaches using the widely applied kernel method [Bous05, BoPe03, JaPrRe08, BoMi10]. In these guises, the reflection principle is often a key element in the solution when the resulting generating function is shown to be D-finite111A function is D-finite if it satisfies a linear differential equation with polynomial coefficients. This is no coincidence: the connection is an expression for the generating function as a diagonal of a rational function. More precisely, in works such as [GeZe92, BoMi10], the analysis results in generating functions expressed as rational sub-series extractions, which can be easily converted to diagonal expressions. Unfortunately, the resulting explicit representations of generating functions can be cumbersome to manipulate. For example, much recent work on walks in Weyl chambers has led to expressions which are determinants of large matrices with Bessel function entries [GrMa93, Grab95, Xin10]. Here, we aim to determine asymptotics for a family of lattice path models arising naturally among those restricted to positive orthants – which correspond to walks in certain Weyl chambers – while avoiding such unwieldly representations. This is acheived by working directly with the diagonal expressions obtained through the recently developed machinery on analytic combinatorics in several variables [PeWi13].

Coupling these techniques – diagonal representation and analytic combinatorics in several variables – yields explicit, yet simple, asymptotic formulas for families of lattice paths. The focus of this article is -dimensional models whose set of allowable steps is symmetric with respect to any axis; we call these models highly symmetric walks. The techniques of analytic combinatorics in several variables apply in a rather straightforward way to derive dominant asymptotics for the number of walks ending anywhere and give an effective procedure to calculate descending terms in the asymptotic expansions. Furthermore, we also consider the subfamily of walks that return to the origin (known as excursions). Once our equations are established, they are suitable input to existing implementations such as that of Raichev [Raic12] (however, in practice one can calculate only the first few terms in these expansions).

The highly symmetric walks we present are amenable to a kernel method treatment. In particular, they fit well into the ongoing study of lattice path classes restricted to an orthant and taking only “small” steps [BoMi10, BoRaSa14]. This collection of models forms a little universe exhibiting many interesting phenomena, and recent work in two and three dimensions has used novel applications of algebra and analysis, along with new computational techniques, to determine exact and asymptotic enumeration formulas. One key predictor of the nature of a model’s generating function (whether it is rational, algebraic, or transcendental D-finite, or none of these) is the order of a group that is associated to each model. This group has its origins in the probabilistic study of random walks, namely [FaIaMa99], and when the group is finite it can sometimes be used to write generating functions as the positive part of an explicit multivariate rational Laurent series. The intimate relation between the generating function of the walks and the nature of the generating function is explored in [BoRaSa14, MeMi14a].

For highly symmetric models in two and three dimensions, this group coincides with that of a Weyl group for walks in the Weyl chambers and , respectively. Indeed, one can use either viewpoint to generalize the study of highly symmetric models to models in arbitrary dimension. As these viewpoints are largely isomorphic, and the kernel method viewpoint is more self-contained, we begin this article by working through a straightforward generalization of the kernel method in order to write the generating function for higher dimensional highly symmetric walks as diagonals of rational functions. We then perform an asymptotic analysis of the coefficients of counting generating functions using techniques from the study of analytic combinatorics in several variables, and consequently link some of the combinatorial symmetries in a walk model to both analytic properties of the generating function and geometric properties of an associated variety. After this is complete, we examine how this connects to the notion of walks in Weyl chambers, use results from their study to determine asymptotic results about excursions, and discuss how the Weyl chamber viewpoint can be used in future work to examine larger classes of lattice path models through diagonals. Next we specify the walks we study in order to precisely state our main results.

1.1. Highly Symmetric Walks

Concretely, the lattice path models we consider are restricted as follows. For a fixed dimension , we define a model by its step set and say that  is symmetric about the axis if implies . We further impose a non-triviality condition: for each coordinate there is at least one step in  which moves in the positive direction of that coordinate (this implies that for each coordinate there is a walk in the model which moves in that coordinate).

The number of walks taking steps in  which are restricted to the positive orthant are studied by expressing the counting generating functions of such models as positive parts of multivariate rational Laurent series, which are then converted to diagonals of rational functions in variables. A first consequence is that all of these models have D-finite generating functions (since D-finite functions are closed under the diagonal operation).

After the above manipulations, these models are very well suited to the asymptotic enumeration methods for diagonals of rational functions outlined in [PeWi13], in particular the cases which were developed by Pemantle, Raichev and Wilson in [PeWi02] and [RaWi08]. Following these methods, we study the singular variety of the denominator of this rational function to determine related asymptotics. The condition of having a symmetry across each axis ensures that the variety is smooth and allows us to calculate the leading asymptotic term explicitly. This is not generally the case, in our experience, and hence we focus on this particular kind of restriction.

Asymptotics     Asymptotics
 
 

Table 1. The four highly symmetric models with unit steps in the quarter plane.

1.2. Main results

We present two main results in this work. The first appears as Theorem 3.4.

Theorem.

Let be a set of unit steps in dimension . If is symmetric with respect to each axis, and takes a positive step in each direction, then the number of walks of length  taking steps in , beginning at the origin, and never leaving the positive orthant has asymptotic expansion

where denotes the number of steps in which have coordinate 1.

This formula is easy to apply to any given model, and for certain infinite families as well.

Example 1.

When there are four non-isomorphic highly symmetric walks in the quarter plane, listed in Table 1. Applying Theorem 3.4 verifies the asymptotic results guessed previously by [BoKa09].

Example 2.

Let , the full set of possible steps. This is symmetric across each axis. We compute that , and for all and so

Example 3.

Let  be the standard basis vector in , and consider the set of steps . Then the number of walks of length taking steps from  and never leaving the positive orthant has asymptotic expansion

The second main result is a comparable statement for excursions, Theorem 7.2.

Theorem.

Let be a set of unit steps in dimension . If is symmetric with respect to each axis, and takes a positive step in each direction, then the number of walks of length taking steps in , beginning and ending at the origin, and never leaving the positive orthant satisfies

1.3. Organization of the paper

The article is organized as follows. Section 2 describes how to express the generating function using an orbit sum by applying the kernel method, following the strategy described in [BoMi10]. We then derive Equation (9), which describes the generating function as the diagonal of a rational power series in multiple variables. Section 3 justifies why the work of Pemantle and Wilson [PeWi13] is applicable, with the asymptotic results computed in Section 3.3. We discuss the sub-dominant growth, and compute an example in Section 5. Section 6 discusses the differential equations satisfied by these generating functions, and how to use creative telescoping techniques to find them. We tabulate some small examples. We conclude with a discussion of how these walks fit into the context of walks in Weyl chambers, which allows us to obtain results on the asymptotics of walk excursions, and also to consider other families of walks.

Determine functional equation for

Represent as the positive part of rational

Convert to a diagonal extraction of

Find the critical points of

Refine to minimal points of

Find asymptotics via formulas of Pemantle and Wilson [PeWi13]

Figure 1. The strategem of determining asymptotics via the generalized kernel method for symmetric walks.

2. Deriving a diagonal expression for the generating function

Fix a dimension and a highly symmetric set of steps . Recall this means that implies . In this section we derive a functional equation for a multivariate generating function, apply the orbit sum method to derive a closed expression related to this generating function, and conclude by writing the univariate counting generating function for the number of walks as the complete diagonal of a rational function.

The following notation is used throughout:

and we write to refer to the ring of Laurent polynomials in the variable .

2.1. A functional equation

To begin, we define the generating function:

(1)

where counts the number of walks of length taking steps from which stay in the positive orthant and end at lattice point . Note that the series is the generating function for the total number of walks in the orthant, and we can recover the series for walks ending on the hyperplane by setting in the series  (the variables are referred to as catalytic variables in the literature, as they are present during the analysis and removed at the end of the ‘reaction’ via specialization to 1). We also define the function (known as either the characteristic polynomial or the inventory of ) by

(2)

In many recent analyses of lattice walks, functional equations are derived by translating the following description of a walk into a generating function equation: a walk is either an empty walk, or a shorter walk followed by a single step. To ensure the condition that the walks remain in the positive orthant, we must not count walks that add a step with a negative -th component to a walk ending on the hyperplane . To account for this, it is sufficient to subtract an appropriate multiple of from the functional equation: , however if a given step has several negative components we must use inclusion and exclusion to prevent over compensation.

This can be made explicit. Let define a -dimensional lattice model restricted to the first orthant, and let  be the generating function for this model, counting the number of walks of length with marked endpoint. Let , so that it is the set of coordinates for which there is at least one step in with in the -th coordinate (this is the full set of indices by our assumptions). Then, by translating the combinatorial recurrence described above, we see that satisfies the functional equation

(3)

Basic manipulations then give the following result.

Lemma 2.1.

Let be the multivariate generating function described above. Then

(4)

for some .

Example 4.

Set . In this case , so vanishes when at least two of the are zero, and the generating function satisfies

2.2. The Orbit Sum Method

The orbit sum method, when it applies, has three main steps: find a suitable group  of rational maps; apply the elements of the group to the functional equation and form a telescoping sum; and (ultimately) represent the generating function of a model as the positive series extraction of an explicit rational function. Bousquet-Mélou and Mishna [BoMi10] illustrate the applicability in the case of lattice walks, and it has been adapted to several dimensions [BoBoKaMe14].

2.2.1. The group

For any -dimensional model, we define the group of rational maps by

(5)

Given , we can consider as a map on through the group action defined by . Due to the symmetry of the step set across each axis, one can verify that always holds. The fact that this group does not depend on the step set of the model – only on the dimension – is crucial to obtaining the general results here. When equals two, the group matches the group used by [FaIaMa99] and [BoMi10]. As we will see in Section 7, corresponds to the Weyl group of the Weyl chamber , where the step set can be studied in the context of Gessel and Zeilberger [GeZe92].

2.2.2. A telescoping sum

Next we apply each of the elements of to Equation (4), and take a weighted sum. Define , where , and let be the map which sends to and fixes all other components of .

Lemma 2.2.

Let be the generating function counting the number of walks of length with marked endpoint. Then, as elements of the ring ,

(6)
Proof.

For each we have and, for the in Equation (4),

Thus, we can apply each to Equation (4) and sum the results, weighted by , to cancel each term on the right hand side. Minor algebraic manipulations, along with the fact that the group elements fix , then give Equation (6). ∎

2.2.3. Positive series extraction

Next, we note that each term in the expansion of

has a negative power of . In fact, except for when is the identity any summand on the left hand side of Equation (6) contains a negative power of at least one variable in any term of its expansion.

With this in mind, for an element we let denote the sum of all terms of which contain only non-negative powers of . Lemma 2.3 then follows from the identity

which can be proven by induction.

Lemma 2.3.

Let be the generating function counting the number of walks of length  with marked endpoint. Then

(7)

where

Since the class of D-finite functions is closed under positive series extraction – as shown in [Li89] – an immediate consequence is the following.

Corollary 2.4.

Under the above conditions on , the generating functions and (thus) are D-finite functions.

2.3. The generating function as a diagonal

Given an element

we let denote the (complete) diagonal operator

There is a natural correspondence between the diagonal operator and extracting the positive part of a multivariate power series, as in Equation (7).

Proposition 2.5.

Let be an element of . Then

(8)
Proof.

Suppose that has the expansion

Then the right hand side of Equation (8) is given by

so that the coefficient of in the diagonal is the sum of all terms with (by assumption there are only finitely many which are non-zero). But this is exactly the coefficient of on the left hand side. ∎

We note also that in the context of lattice path models with step set , the modified generating function is actually a power series in the variables (as a walk cannot move farther on the integer lattice than its number of steps). Combining Lemma 2.3 and Proposition 2.5 implies that the generating function for the number of walks can be represented as , where

(9)

To be precise, and are defined as the numerator and denominator of Equation (9).

Example 5.

For the walks defined by , we have

Note that this rational function is not unique, in the sense that there are other rational functions whose diagonals yield the same counting sequence.

2.4. The singular variety associated to the kernel

Here, we pause to note that the combinatorial symmetries of the step sets that we consider affect the geometry of the variety of – called the singular variety. This has a direct impact on both the asymptotics of the counting sequence under consideration and the ease with which its asymptotics are computed. In particular, any factors of the form present in the denominator of this rational function before simplification could have given rise to non-simple poles and thus made the singular variety non-smooth. Although non-smooth varieties can be handled in many cases – see [PeWi13] – having a smooth singular variety is the easiest situation in which one can work in the multivariate setting. Understanding the interplay between the step set symmetry and the singular variety geometry, and in the process dealing with the non-smooth cases, is promising future work.

3. Analytic combinatorics in several variables

Following the work of Pemantle and Wilson [PeWi02] and Raichev and Wilson [RaWi08], we can determine the dominant asymptotics for the diagonal of the multivariate power series by studying the variety (complex set of zeroes) of the denominator

To begin, a particular set of singular points – called the critical points – containing all singular points which could affect the asymptotics of are computed in Section 3.1. The set of critical points is then refined to those which determine the dominant asymptotics up to an exponential decay in Section 3.2; this refined set is called the set of minimal points as they are the critical points which are ‘closest’ to the origin in a sense made precise below. The enumerative results come from calculating a Cauchy residue type integral, and after determining the minimal points we determine asymptotics in Section 3.3 using pre-computed formulas for such integrals which can be found in [PeWi13]. In fact, up to polynomial decay there is only one singular point which determines dominant asymptotics for each model – the point – and this uniformity aids greatly in computing the quantities required in the analysis of a general step set, in order to obtain Theorem 3.4.

We first verify our claim in the previous section that the variety is smooth (that is, at every point on one of the partial derivatives or does not vanish). Indeed, any non-smooth point on would have to satisfy both

which can never occur. Equivalently, this shows that at each point in there exists a neighbourhood such that is a complex submanifold of .

3.1. Critical points

The next step is to find the critical points. Determined through an appeal to stratified Morse theory, for a smooth variety the critical points are precisely those which satisfy the following critical point equations:

which we now solve. Given , define

As each step in  has coordinates taking values in , we may collect the coefficients of the variable, and use the symmetries present to write

(10)

which uniquely defines the Laurent polynomials and . With this notation the equation becomes

which implies

(11)

Note that while is a polynomial, itself is a Laurent polynomial, so one must be careful when specializing variables to 0 in the expression. This calculation characterizes the critical points of .

Proposition 3.1.

The point is a critical point of if and only if for each either:

  1. or,

  2. the polynomial has a root at .

Proof.

We have shown above that the critical point equations reduce to Equation (11). Furthermore, if  were zero at a point on then , a contradiction. ∎

It is interesting to note that the polynomial has combinatorial signifigance, as the subset of which encodes only the steps which move forwards in their coordinate.

3.2. Minimal points

Among the critical points, only those which are ‘closest’ to the origin will contribute to the asymptotics, up to an exponentially decaying error. This is analogous to the single variable case, where the singularities of minimum modulus are those which contribute to the dominant asymptotic term. To be precise, for any point we define the closed polydisk

The critical point is called strictly minimal if , and finitely minimal if the intersection contains only a finite number of points, all of which are on the boundary of . Finally, we call a critical point isolated if there exists a neighbourhood of where it is the only critical point. In our case, we need only be concerned with isolated finitely minimal points.

Proposition 3.2.

The point is a finitely minimal point of the variety . Furthermore, any point in is an isolated critical point.

Proof.

The point is critical as it lies on and its first coordinates are all one. Suppose lies in , where we note that any choice of uniquely determines on . Then, as ,

But implies for each . Thus, the above inequality states that the sum of complex numbers of modulus at most one has modulus . The only way this can occur is if each term in the sum has modulus one, and all terms point in the same direction in the complex plane. By symmetry, and the assumption that we take a positive step in each direction, there are two terms of the form and in the sum, so that must be 1 in order for them to point in the same direction. This shows , and the same argument applies to each , so there are at most points in .

By Proposition 3.1 every such point is critical, and to show it is isolated it is sufficient to prove for all . Indeed, if then implies

by our assumption that contains a step which moves forward in the coordinate. This contradicts . ∎

3.3. Asymptotics Results

To apply the formulas of [PeWi02] we need to define a few quantities. To start, we note that on all of we may parametrize the coordinate as

For each point , the analysis of [PeWi02] shows that the asymptotics of the integral in question which determines asymptotics for a given model depends on the function

(12)

Let denote the determinant of the Hessian of at :

if then we say is non-degenerate. The main asymptotic result of smooth multivaritate analytic combinatorics, in this restricted context, is the following (the original result allows for asymptotic expansions of coefficient sequences more generally defined from multivariate functions than the diagonal sequence).

Theorem 3.3 (Adapted from Theorem 3.5 of [PeWi02]).

Suppose that the meromorphic function has an isolated strictly minimal simple pole at . If does not vanish at then there is an asymptotic expansion

(13)

for constants , where is the degree to which vanishes near . When does not vanish at then and the leading term of this expansion is

(14)

In fact, Corollary 3.7 of [PeWi02] shows that in the case of a finitely minimal point one can simply sum the contributions of each point. Combining this with the above calculations gives our main result.

Theorem 3.4.

Let be a set of unit steps in dimension . If is symmetric with respect to each axis, and takes a positive step in each direction, then the number of walks of length taking steps in , beginning at the origin, and never leaving the positive orthant has asymptotic expansion

(15)

where denotes the number of steps in which have coordinate 1.

Proof.

We begin by verifying that each point satisfies the conditions of Theorem 3.3:

1. is a simple pole:

As is smooth, the point is a simple pole.

2. is isolated:

This is proven in Proposition 3.2.

3. does not vanish at :

This follows from .

4. is non-degenerate:

Directly taking partial derivatives in Equation (12) implies

Since we see that . Similarly, one can calculate that and for , so that the Hessian of at is a diagonal matrix and

(16)

The proof of Proposition 3.2 implies that for any , so each is non-degenerate.

Thus, we can apply Corollary 3.7 of [PeWi02] and sum the expansions (13) at each point in to obtain the asymptotic expansion

(17)

for constants , where is the degree to which vanishes near . Since the numerator vanishes at all points of except for , the dominant term of (17) is determined only by the contribution of . Substituting the value for given by Equation (16) into Equation (14) gives the desired asymptotic result. ∎

4. Examples

We now give two examples, both of which calculate critical points by directly solving the critical point equations. The first example has only a finite number of critical points, all of which are minimal points. In contrast, the second example contains a curve of critical points (however, as guaranteed by Proposition 3.2, no points on this curve are minimal points).

Example 6.

Consider the model in three dimensions restricted to the positive octant taking the eight steps

The kernel equation here is

with characteristic polynomial

The generalized orbit sum method implies where