Some exact asymptotics in the counting of walks in the quarter-plane

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by  Guy Fayolle

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Some exact asymptotics
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RESEARCH R
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N° 7863 63
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January2012 02
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Project-TeamImara S
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Some exact asymptotics in the counting of
walks in the quarter-plane
Guy Fayolle∗, Kilian Raschel †
Project-Team Imara
Research Report n° 7863 — January 2012 — 17 pages
Abstract: Enumeration of planar lattice walks is a classical topic in combinatorics, at
the cross-roads of several domains (e.g., probability, statistical physics, computer science).
The aim of this paper is to propose a new approach to obtain some exact asymptotics for
walks confined to the quarter plane.
Key-words: Boundaryvalueproblem,generating function,randomwalk in thequarter
plane, singularity analysis.
∗ INRIAParis-Rocquencourt,DomainedeVoluceau,BP105,78153LeChesnayCedex,France. Email:
[email protected]
†LaboratoiredeMath´ematiquesetPhysiqueTh´eorique,Universit´eFranc¸oisRabelais,F´ed´erationDenis
Poisson-CNRS, Parc deGrandmont, 37200 Tours, France. Email: [email protected]
RESEARCHCENTRE
PARIS–ROCQUENCOURT
DomainedeVoluceau,-Rocquencourt
B.P.105-78153LeChesnayCedex
Quelques estimations asymptotiques exactes pour le
comptage de marches dans le quart de plan
R´esum´e : L’´enum´eration de marches planaires sur des treillis est un domaine classique
de la combinatoire, `a la crois´ee de plusieurs disiplines (probabilit´es, physique statistique,
informatique). Cet article propose une nouvelle m´ethode donnant des estimations asymp-
totiques exactes pour certaines marches confin´ees dans le quart de plan.
Mots-cl´es : Analyse de singularit´e, fonction g´en´eratrice, marche al´eatoire dans le quart
de plan, probl`eme aux limites.
Some exact asymptotics in the counting of walks 3
Fig. 1.1: Four famous examples, known as the simple, Kreweras’, Gessel’s and Gouyou-
Beauchamps’ walks, respectively
1 Introduction
Enumeration of planar lattice walks is a most classical topic in combinatorics. For a given
set of admissible steps (or jumps), it is a matter of counting the number of paths of a
S
certain length, which start and end at some arbitrary points, and might even be restricted
to some region of the plane. Then three natural important questions arise.
(i) How many such paths do exist?
(ii) What is the asymptotic behavior, as their length goes to infinity, of the number of
walks ending at some given point or domain (for instance one axis)?
(iii) What is the nature of the generating function of the numbers of walks? Is it holo-
nomic§, and, in that case, algebraic or even rational?
If the paths are not restricted to a region, or if they are constrained to remain in a half-
plane, it turns out [6] that the counting function has an explicit form (question (i)), and
is, respectively, rational or algebraic (question (iii)). Question (ii) can then besolved from
the answer to (i).
The situation happens to be much richer if the walks are confined to the quarter plane
Z2. As an illustration, let us recall that some walks admit an algebraic counting function,
+
see [14] for Kreweras’ walk (see Figure 1.1), while others admit a counting function which
is not even holonomic, see [6] for the so-called knight walk.
In the sequel, we focus on walks confined to Z2, starting at the origin and having small
+
steps. This means exactly that the set of admissible steps is included in the set of the
S
eight nearest neighbors, i.e., 1,0,1 2 (0,0) . By using the classical Kronecker’s
S ⊂ {− } \{ }
delta, we shall write
1 if (i,j) ,
δi,j = 0 if (i,j) ∈/ S. (1.1)
(cid:26) ∈S
On the boundary of the quarter plane, allowed jumps are the natural ones: steps that
would take the walk out Z2 are obviously discarded, see examples on Figure 1.1.
+
There are 28 such models. Starting from the existence of simple geometrical symmetries,
Bousquet-M´elou and Mishna [5] have shown that there are in fact 79 types of essentially
distinct walks—we will often refer to these 79 walks tabulated in [5]. For each of these
RR n°7863
4 G. Fayolle & K.Raschel
79 models, q(i,j,n) will denote the number of paths confined to Z2, starting at (0,0) and
+
ending at (i,j) after n steps. The associated generating function will be written
Q(x,y,z) = q(i,j,n)xiyjzn. (1.2)
i,j,n>0
X
Answerstoquestions (i)and(iii) have beenrecently determinedfor all79models ofwalks.
To present these results, we need to define a certain group, introduced in a probabilistic
context in [18], and called the group of the walk. This group W = Ψ,Φ of birational
h i
transformations in C2 leaves invariant the function δ xiyj, and has the two
−16i,j61 i,j
following generators:
P
−16i61δi,−1xi 1 −16j61δ−1,jyj 1
Ψ(x,y) = x, , Φ(x,y) = ,y . (1.3)
δ xiy δ yj x
P−16i61 i,+1 ! P−16j61 +1,j !
Clearly Ψ Ψ = ΦPΦ = id, and W is a dihedral group,Pwhose order is even and at least
◦ ◦
four. The order of W is calculated in [5] for each of the 79 cases: 23 walks admit a finite
group (then of order 4,6 or 8), and the 56 others have an infinite group.
The expression of the counting function Q(x,y,z)—or equivalently the answer to (i)—for
the 23 walks with a finite group has been determined in [2, 5]. It was basically derived
by applying a kernel method (see, e.g., [13]) to the functional equation (2.1) satisfied by
Q(x,y,z).
Asforthe56modelswithaninfinitegroup,thefunctionQ(x,y,z)relatedtothe5singular
walks (i.e., walks having no jumps to the West, South-West and South) was found in
[19, 20] by using elementary manipulations on (2.1). For the remaining 51 non-singular
walks, Q(x,y,z) was found in [21] by the method, initiated in [8, 9], of reduction to
boundary value problems (BVP).
Concerning (iii), the answer was obtained in [2, 5, 10] for the 23 walks with a finite group:
the function Q(x,y,z) is always holonomic, and even algebraic when ijδ is
−16i,j61 i,j
positive. For the 56 models with an infinite group, it was also proved that they all admit
P
a non-holonomic counting function: see [19, 20] for the 5 singular walks, and [16] for the
non-singular ones.
Let us now focus on question (ii), which actually is the main topic of this paper. A
priori, it has a link with question (i) (indeed, being provided with an expression for the
generatingfunction,oneshouldhopefullybeabletodeduceasymptoticsofitscoefficients),
and with (iii) as well (since holonomy is strongly related to singularities, and hence to the
asymptotics of the coefficients). As we shall see, the situation is not so simple, notably
because the expression of the counting function (1.2) is fairly complicated to deal with, in
particular whenitis given underanintegral form, whichis afrequentsituation inconcrete
case studies.
§A function of several complex variables is said to be holonomic if the vector space over the field of
rationalfunctionsspannedbythesetofallderivativesisfinitedimensional. Inthecaseofonevariable,this
istantamounttosayingthatthefunctionissolutionofalineardifferentialequationwherethecoefficients
are rational functions (see [13]).
Inria
Some exact asymptotics in the counting of walks 5
If we restrict question (ii) to the computation of asymptotic estimates for the number of
walks endingat theorigin (0,0) (these numbersare thecoefficients of Q(0,0,z), see (1.2)),
a very recent paper [7] solved the problem for much more general random walks in Rd, for
any d 1: indeed, the authors obtain the leading term of the estimate, up to a coefficient
≥
involving two unknown harmonic functions. On the other hand, it is impossible to deduce
from [7] the asymptotic behavior of the number of walks ending, for instance, at one axis
(coefficients of Q(1,0,z) and Q(0,1,z)), or anywhere (coefficients of Q(1,1,z)).
Still with respect to question (ii), it is worth mentioning several interesting conjectures by
Bostan and Kauers [3, 4], which are proposed independently of the finiteness of the group.
The aim of this paper is to propose a uniform approach to answer question (ii), which, as
weshallsee, worksforboth finiteorinfinitegroups,andforwalks notnecessarily restricted
to have closed trajectories as in [7].
First,Section2presentsthebasicfunctionalequation,whichisthekeystoneoftheanalysis.
Then, Section 3 illustrates our approach by considering the simple walk (the jumps of
which are represented in Figure 1.1), which has a group of order 4. In this case, we obtain
the explicit expression of Q(x,y,z), allowing to get exact asymptotics of the coefficients
of Q(0,0,z), Q(1,0,z) (and by an obvious symmetry, also of Q(0,1,z)) and of Q(1,1,z),
by making a direct singularity analysis. Section 4 sketches the main differences (in the
analytic treatment) occurringbetween the simple walk and any of the 78 other models. In
particular, we explain the key phenomena leading to singularities. Indeed, a singularity is
due to the fact that a certain Riemann surface sees its genus passing from 1 to 0, while
another singularity takes place at z = 1/ , where denotes the number of possible
|S| |S|
steps of the model. Also, as in the probabilistic context (although this will not be shown
in this paper), the orientation of the drift vector has a clear impact on the nature of the
singularities which lead to the asymptotics of interest.
To conclude (!) this introduction, let us mention that our work has clearly direct con-
nections with probability theory. Indeed, with regard to random walks evolving in the
quarter plane, and more generally in cones, the question of the asymptotic tail distribu-
tion of the first hitting time of the boundary has been for many decades of great interest.
The approach proposed in this paper should hopefully lead to a complete solution of this
problem for random walks with jumps to the eight nearest neighbors.
2 The basic functional equation
A common starting point to study the 79 walks presented in the introduction is to estab-
lish the following functional equation (proved in [5]), satisfied by the generating function
defined in (1.2).
K(x,y,z)Q(x,y,z) = c(x)Q(x,0,z)+c(y)Q(0,y,z) δ Q(0,0,z) xy/z, (2.1)
−1,−1
− −
where c and c are defined in (2.3) and
e
K(x,y,z) = xy[ δ xiyj 1/z]. (2.2)
e −16i,j61 i,j −
P
RR n°7863
6 G. Fayolle & K.Raschel
Equation (2.1) holds at least in x 6 1, y 6 1, z < 1/ , since obviously q(i,j,n) 6
n. In (2.1), we note that the{δ| |’s defi|ne|d in (|1.|1) pla|yS|t}he role (up to a normalizing
i,j
|S|
condition) of the usual transition probabilities p ’s in a probabilistic context.
i,j
Our main goal is to analyze the dependency of Q(x,y,z) with respect to the time variable
z, which in fact plays merely the role of a parameter, as far as the functional equation
(2.1) is concerned. Remark that Q(x,y,z) has real positive coefficients in its power series
expansion. Thus, letting R denote the radius of convergence, say of Q(1,0,z), it is worth
noting that, by Pringsheim’s theorem (see [23]), z = R is a singular point of this function.
Thequantity (2.2)thatappearsin(2.1)iscalledthekernelofthewalk. Itcanberewritten
as
a(y)x2+[b(y) y/z]x+c(y) = a(x)y2+[b(x) x/z]y+c(x),
− −
where
e e e
a(y) = δ yj, b(y) = δ yj, c(y) = δ yj,
−16j61 1,j −16j61 0,j −16j61 −1,j (2.3)
a(x) = δ xi, b(x) = δ xi, c(x) = δ xi.
P−16i61 i,1 P−16i61 i,0 P−16i61 i,−1
e e e
Let us also dePfine P P
d(y,z) = [b(y) y/z]2 4a(y)c(y), d(x,z) = [b(x) x/z]2 4a(x)c(x). (2.4)
− − − −
Fromenow on, wee shall take z teo beea real variable. Indeed this is in noway a restriction,
as it will emerge from an analytic continuation argument.
For z (0,1/ ), the polynomial d (resp. d) has four roots (one at most being possibly
∈ |S|
infinite) satisfying in the x-plane (resp. y-plane)
e
x (z) < x (z) < 1 < x (z) < x (z) 6 , y (z) < y (z) < 1 < y (z) < y (z) 6 ,
1 2 3 4 1 2 3 4
| | | | ∞ | | | | ∞
(2.5)
as shown in [21]. Then consider the algebraic functions X(y,z) and Y(x,z) defined by
K(X(y,z),y,z) = K(x,Y(x,z),z) = 0.
With the notations (2.3) and (2.4), we have
b(y,z) d(y,z) b(x,z) d(x,z)
X(y,z) = − ± , Y(x,z) = − ± . (2.6)
2a(y,qz) 2a(x,z)
p
e e
From now on, we shall denote by X ,X (resp. Y ,Y ) the two branches of these algebraic
0 1 0 1
e
functions. They can be separated (see [9]), to ensure X 6 X , resp. Y 6 Y , in the
0 1 0 1
| | | | | | | |
whole complex plane C.
3 The example of the simple walk
To illustrate the method announced in the introduction, we consider in this section the
simple walk (i.e., δ = 1 only for couples (i,j) such that ij = 0, see Figure 1.1). Then
i,j
the main following results hold.
Inria
Some exact asymptotics in the counting of walks 7
Proposition 3.1. For the simple walk,
1 1 1 2uz (1 2uz)2 4z2
Q(0,0,z) = − − − − 1 u2du. (3.1)
π z2 −
Z−1 p
p
Note that Q(0,0,z) counts the numbers of excursions starting from (0,0) and returning
to (0,0).
Proposition 3.2. For the simple walk, as n ,
→ ∞
416n
q(0,0,2n) .
∼ π n3
Proposition 3.3. For the simple walk,
1 1 1 2uz (1 2uz)2 4z2 1+u
Q(1,0,z) = − − − − du.
2π z2 1 u
Z−1 p r −
The generating function Q(1,0,z) depicts the numbers of walks starting from (0,0) and
endingwhentheyreachthehorizontalaxis. Inaddition,byanevidentsymmetry,Q(0,1,z) =
Q(1,0,z).
Proposition 3.4. For the simple walk, as n ,
→ ∞
8 4n
q(i,0,n) .
∼ πn2
i>0
X
Having the expressions of Q(1,0,z) and Q(0,1,z), the series Q(1,1,z) is directly obtained
from (2.1). The asymptotics of its coefficients, which represent the total number of walks
of a given length, is the subject of the next result.
Proposition 3.5. For the simple walk, as n ,
→ ∞
44n
q(i,j,n) .
∼ π n
i,j>0
X
Proof of Proposition 3.1. The key point is to reduce the computation of Q(x,0,z) to a
BVP set on Γ = t C : t = 1 , according to the procedure proposed in [8, 9]. Indeed,
{ ∈ | | }
for any t Γ, we have
∈
tY (t,z) t¯Y (t¯,z)
c(t)Q(t,0,z) c(t¯)Q(t¯,0,z) = 0 − 0 , (3.2)
− z
where t¯stands for the complex conjugate of t. The proof of (3.2) relies on simple ma-
nipulations on the functional equation (2.1). The main argument is the following. On
K(x,y,z) = 0, letting y tend successively from above (y+) and from below (y−) to an
arbitrary point y on the cut [y (z),y (z)] defined in (2.5), Q(0,y,z) remains continuous,
1 2
and thus can be eliminated by computing the difference Q(0,y+,z) Q(0,y−,z) = 0 in
−
the functional equation (2.1), see [8, 9] for full details.
RR n°7863
8 G. Fayolle & K.Raschel
In the present case, a pleasant point (from a computational point of view) is that, on the
unit circle, we have t¯= 1/t. Hence, after mutiplying both sides of (3.2) by 1/(t x) with
−
x < 1, integrating over Γ, and making use of Cauchy’s formula, we get the following
| |
integral form (noting that here c(x) = x)
1 tY (t,z) t¯Y (t¯,z)
0 0
c(x)Q(x,0,z) = − dt, x < 1. (3.3)
2iπz t x ∀| |
ZΓ −
For t Γ, we know from general results [9, 21] that for z [0,1/4], Y (t,z) is real and
0
∈ ∈
belongs to the segment [y (z),y (z)], the extremities of which are the two branch points
1 2
located inside the unit circle in the y-plane, see Section 2. Then it is not difficult to check
that the integral in the right-hand side of (3.3) does vanish at x = 0. Hence we can
entitled to write
1 tY (t,z) t¯Y (t¯,z) 1 tY (t,z) t¯Y (t¯,z)
0 0 0 0
Q(0,0,z) = lim − dt = − dt, (3.4)
x→02iπxz ZΓ t−x 2iπz ZΓ t2
where the last inequality follows from l’Hospital’s rule.
To work on formula (3.4), it is convenient to make the straightforward change of variable
t = eiθ, which yields
i 2π 1 2π
Q(0,0,z) = sinθe−iθY (eiθ,z)dθ = sin2θY (eiθ,z)dθ, (3.5)
0 0
2πz πz
Z0 Z0
wherewehaveusedthefactthatY (eiθ,z)isrealandY (eiθ,z) = Y (e−iθ,z). Furthermore,
0 0 0
we have by (2.6)
1 2zcosθ (1 2zcosθ)2 4z2
Y (eiθ,z) = − − − − . (3.6)
0
2z
p
Then, instantiating (3.6) in (3.5), we get after some light algebra
1 1 π
Q(0,0,z) = sin2θ (1 2zcosθ)2 4z2dθ. (3.7)
2z2 − πz2 − −
Z0
p
Putting now u= cosθ in the integrand, equation (3.7) becomes exactly (3.1).
Proof of Proposition 3.2. The function Q(0,0,z) is holomorphic in C (( , 1/4]
\ −∞ − ∪
[1/4, )), as it easily emerges from Proposition 3.1. Accordingly, the asymptotics of
∞
its coefficients will be derived from the behavior of the function in the neighborhood of
1/4 (see, e.g., [13]). On the other hand Q(0,0,z) is even, as can be been in (3.1), or
±
directly since q(0,0,2n+1) = 0, for any n > 0, in the case of the simple walk (see Figure
1.1). Consequently, it suffices to focus on the point 1/4.
First,werewrite (1 u2)[(1 2uz)2 4z2]as (1 u)(1 2(u+1)z) (1+u)(1 2(u 1)z),
− − − − − − −
where the second radical admits the expansion µ (u 1)i(z 1/4)j. Therefore,
p p i,j>0 i,j − − p
for z in a neighborhood of 1/4, we have
P
1 1
Q(0,0,z) = µ (z 1/4)j (u 1)i (1 u)(1 2(u+1)z)du. (3.8)
−πz2 i,j − − − −
i,j>0 Z−1
X p
Inria