Combinatorics meets potential theory

Using potential theoretic techniques, we show how it is possible to determine the dominant asymptotics for the number of walks of length n, restricted to the positive quadrant and taking unit steps in a “balanced” set Γ. The approach is illustrated through an example of inhomogeneous space walk. This walk takes its steps in {←, ↑,→, ↓} or {↙,←,↖, ↑,↗,→,↘, ↓}, depending on the parity of the coordinates of its positions. The exponential growth of our model is (4φ)n, where φ = 1+ √ 5 2 denotes the Golden ratio, while the subexponential growth is like 1/n. As an application of our approach we prove the non-D-finiteness in two dimensions of the length generating functions corresponding to nonsingular small step sets with an infinite group and zero-drift.

Given a set Γ of allowed steps, the basic enumerative question is to determine the number of walks confined to Q, starting from (x, y) ∈ Q, having length n = 1, 2, . . . the electronic journal of combinatorics 23(2) (2016), #P2.28 and taking steps in Γ only.A walk in which the set of allowable steps Γ is contained in { , ←, , ↑, , →, , ↓} is said to have small steps.
The enumeration of small steps walks restricted to the positive quadrant has focused primarily on the associated generating functions.In contrast with the corresponding 1dimensional problem, where the generating functions are always algebraic [2,9,14], in the 2-dimensional case, depending on the choice of the set Γ, the generating functions may be algebraic, D-finite and some times not even D-finite [8,10,12].
Three fundamental works should be mentioned.Firstly, the seminal work of Bousquet-Mélou and Mishna [11] (see also [33]) initiating a complete classification of small steps walks in Q and showing that the nature of the generating function is correlated to the finiteness of a certain group of 2-dimensional transformations associated with Γ. Bousquet-Mélou and Mishna proved that among the 256 possible small-step walks, there are exactly 79 different models (up to symmetries) and 23 have a finite group.Among the 23, they proved that 22 models have D-finite generating functions (the 23rd was shown algebraic in [4]).Secondly, the Kurkova and Raschel result [27] proving a conjecture of Bousquet-Mélou and Mishna about the non-D-finiteness of the trivariate generating function which takes into account the length of the walk and the position of its endpoint for the 51 nonsingular models among the 56 with infinite groups.The remaining 5 models were shown to have non-D-finite length generating functions by Mishna and Rechnitzer [34] and Melczer and Mishna [31].Thirdly, the Bostan et al. result [5] establishing the non-D-finiteness of excursions generating functions for the same 51 nonsingular walks (the nature of the length generating function for all quadrant walks is still unknown in those 51 cases).
After the important progress made on understanding small steps quadrant walks, efforts are now deployed to move one dimension higher (small steps walks in octants [6]) and to address walks with big jumps (see [12,18]).
A natural question about quadrant walks that has not received any special attention despite its interest provided the first motivation for this paper.What does happen if we let the set where the walk takes its steps depend on its positions?
At first glance, it might seem difficult to investigate inhomogeneous space walks because of the great difficulty of analysis of their generating functions [30].A further major difficulty that appears in the non-homogeneous case is that paths are no longer equally probable and it is no longer possible to use the counting formula relating the enumeration problem to the survival probability in the quadrant to count them (see [13]).
Our aim is to offer an alternative approach allowing us to investigate quadrant walks.To explain the underlying principle we choose to present this approach through an example.However, our method has a more general scope and applies in higher dimension and for models with longer steps.It relies on a systematic use of tools and techniques from discrete potential theory developed in [24,35,36].It assumes a centering condition and requires the construction of an appropriate harmonic function whose existence is difficult to establish in full generality.Nevertheless, if we restrict ourselves to the homogeneous case, the method applies quite generally and offer an alternative to the probabilistic approach.This allows us to answer some open questions in the subject.We show in particular in §5, that for the three models with nonsingular small step set, zero-drift and infinite group (among the 51 nonsingular models considered in [5]) the length generating function is not D-finite.

Model description and main result
We consider walks that starts at (x, y) ∈ Q and take their steps w = w 1 , w 2 , . . . in variable multi-sets Γ ⊂ { , ←, , ↑, , →, , ↓} which depend on the position of W j = j i=1 w i .We denote by Q(x, y) the set of all walks that stay in Q and by Q n (x, y) ⊂ Q(x, y) the subset of walks of length n.The main quantity we investigate is ν n (x, y) = |Q n (x, y)|.
Figure 2: Two golden paths among the 238525020 14-steps quadrant paths, starting from (1, 1) and ending at (3,6).The path corresponding to the solid line is much more likely than the dotted one (it has probability 2 −36 vs 2 −40 for the dotted path).
We use the notation F x,y to highlight the dependence of the generating function on the starting point (x, y).It is the will to understand this dependence that explains our formulation of the counting problem.
As ν n (x, y; x , y ) = 0 as soon as |x − x | > n or |y − y | > n, the inner sum in ( 3) is finite, and F x,y (t; ξ, η) can be identified with a power series in t with coefficients in Q[ξ, η].Choosing ξ = η = 1 yields a power series whose coefficients count the quadrant walks with prescribed number of steps.In the classical theory the growth of the coefficients of the generating functions is related to the location and nature of their singularities [19,20].The generating function methodology, however, seems difficult to implement in the case of spatially inhomogeneous walks, the recurrences in (1) being built with "variable" coefficients.
In order to make a first step towards understanding spatially inhomogeneous walks we investigate a new model of walks, taking their steps w = w 1 , w 2 , . . .w j , . . .alternately in one of the two multi-set Γ even = {←, ↑, →, ↓} ⊂ Γ odd = { , ←, , ↑, , →, , ↓} according to the following rule: if W j = j i=1 w i = (x, x + 2k), k ∈ Z, then the step w j+1 is taken from Γ even and it is taken from Γ odd otherwise (see Figure 2).We call them golden walks.
It is well known that the number of n-steps walks on the square lattice, with ←, ↑, → and ↓ steps (this is the so-called Pólya walk) that start from the origin and stay in the first quadrant grows like 4 n /n.On the other hand, the number of n-steps quadrant walks taking their steps in the full set { , ←, , ↑, , →, , ↓} (the so-called King walk) grows like 8 n /n.In our model the Pólya walk is modified so that it is allowed to move from the points whose coordinates are of opposite parity in the four additional directions , , and .How does the asymptotics of walks of length n, restricted to the positive quadrant, change?
The answer is given by denotes the Golden ratio.
The following remarks may be helpful in placing the above theorem in the right perspective.
(i) As we are interested mainly by the asymptotic behavior of ν n (x, y), n → ∞, the condition x, y c √ n is not really restrictive.(ii) The factor (4φ) n that appears in ( 4) is related to the number of unrestricted paths of length n (see §3.2.below).The same factor appears in the estimates of the number of n-steps paths confined to the half-space H = {(x, y) ∈ Z 2 , y 0} and starting at (x, y) ∈ H.Note this number ν H n (x, y).Our method allows to show that (iii) It is important to clarify the connection between the functions u(x, y) = xy, u H (x, y) = y and the factors 1/n and 1/ √ n that appear in the lower and upper bounds of ( 4) and ( 5).It will be shown in §3.1 below that the function u satisfies a discrete heat type equation (see Eq. ( 9) below).Moreover, it is positive and vanishes on ∂Q.The same kind of verification is easily done with u H which vanishes on ∂H.The existence and uniqueness of discrete harmonic functions was recently established, in a related context, for spatially inhomogeneous random walks on orthants in Z d [7].The same reasoning used in [7], based on an analog of Theorem 3 below, allows to show in the case here considered the uniqueness (up to multiplicative constants) of time-independent positive solutions of Eq. ( 9) vanishing on ∂Q.In terms of the function u (resp.u H ) the factor 1/n (resp.1/ √ n) can be interpreted as and the quotient xy/n (resp.y/n) as What counts in the choice of the point ( (iv) Due to the previous considerations, it is natural to divide ν n (x, y) by (4φ) n in (4) and try to find a similar interpretation for the resulting quotient in terms of a positive solution of (9) normalized by its value at a point located at a distance ≈ √ n from ∂Q.This can be achieved by considering unrestricted paths.

The heat type equation and the unrestricted paths
Let N n (x, y) denote the number of unconstrained walks starting at (x, y) and of length n ∈ N. We set N n (x, y) = 1 for n = 0 and denote by 1 Γ(x,y) the characteristic function of the set Γ(x, y).N n (x, y) satisfies the equation Dividing (1) by N n+1 (x, y) we see that U satisfies and using (6) we can rewrite this equation Introducing the discrete time and space derivatives we see that U(n; (x, y)) satisfies the following discrete heat type equation: the electronic journal of combinatorics 23(2) (2016), #P2.28

The harmonic function u.
Let us show that the function u : Q → R, (x, y) → u(x, y) = xy satisfies equation ( 9) in the case of golden walks.As u is independent of n the left-hand side is zero.To compute the right-hand side we first observe that u(x + h, y + k) − u(x, y) = xk + yh + hk.This implies that This shows that for u to be solution of ( 9) requires (see ( 8))

Counting the unrestricted paths.
Let N n (x, y) denote the number of unrestricted walks starting at (x, y) and of length n ∈ N. We have N n (x, y) = N o n (x, y) + N e n (x, y), where N o n (x, y) (resp.N e n (x, y)) denote the number of walks ending on odd sites, i.e. sites (x , y ) such that y − x ≡ 1 mod(2) (resp.on even sites).Since every odd site is accessible from its eight neighbors, we have (11) N o n+1 (x, y) = 4N o n (x, y) + 4N e n (x, y).
In contrast, even sites are accessible only from their four odd neighbors.This implies that ( 12) Adding ( 11) and ( 12) we obtain which shows that (F n (x, y)) is a Fibonacci sequence.Using the initial conditions the electronic journal of combinatorics 23(2) (2016), #P2.28 we deduce the following formulas: where and F n denotes the n th Fibonacci number; the second equality in ( 13) and ( 14) results from the Binet's Fibonacci number formula.
On the other hand, using ( 8), ( 13) and ( 14) we deduce that in the case where y − x ≡ 0 mod(2), and that in the case where y − x ≡ 1 mod(2).We will retain from the previous calculations, the following facts: for appropriate C, α > 0.

Towards the proof of Theorem 1
It follows from (15) that Using (7) and the obvious inequality U(n; (x, y)) 1; we see then that in order to establish the upper bound of Theorem 1, it is sufficient to show that For the lower estimate we will need to reverse inequality (17) and a lower bound for appropriate c, C > 0. It is discrete potential theory that will allow us to establish (17), its reverse and the lower bound (18).
the electronic journal of combinatorics 23(2) (2016), #P2.28 4 Discrete potential theory and Proof of Theorem 1 Ideas from potential theory, in particular, harmonic functions, maximum principle, Harnack inequalities and their parabolic and boundary variants have strong connections with our problem.In fact general concepts of potential theory, have been borrowed since a long time by discrete probability theory [15,39].This proved very useful and successful for random walk analysis [28,29].While it is beyond the scope of this paper to explain in detail the related concepts, we will try to explain the main tools that are relevant to our problem, i.e.Harnack inequalities.For this we need to introduce some notation.Two points in Z 2 will be said to be adjacent if the distance between them is unity.A subset A ⊂ Z 2 of cardinality |A| 2 will be called connected if for any two points of A there is a path consisting of segments of unit length connecting them in such a manner that the end points of these segments are all in A. A set of points is a domain if it is connected.The symbol A will be used in the following to denote a domain of Z 2 .
Given a map Γ : (s, t) ∈ Z 2 → Γ(s, t) ⊂ Z 2 , we define the boundary ∂ Γ A of A (with respect to Γ) by We shall assume that Γ satisfies: (19) There exists The closure of A will be denoted by A and defined by Let Γ satisfying (19) and Π : and ( 21) We shall also assume that Π is strongly aperiodic.This means that, given any (x, y) ∈ Z 2 , there exists some integer n 0 = n 0 (x, y) such that Π(n; (0, 0); (x, y))) > 0 for all n n 0 .
The concept of a Π-caloric function generalizes the notion of discrete harmonic function.In particular, caloric functions satisfy an adapted version of the maximum principle.More precisely, a caloric function on a finite set B attains its maximum in B on the lateral boundary ∂ l B. Another basic property of harmonic functions is Harnack principle, which states that a positive harmonic function on a ball of radius R is roughly constant on the ball with the same center and radius R/2.Harnack principle generalizes for nonnegative Π-caloric as follows.
Kuo and Trudinger proved their Harnack principle in the setting of implicit difference schemes.To see how the explicit case, which corresponds to our Theorem 2, can be obtained by their method one can proceed as follows.Start with [ [24], §2] and replace it by [ [25], §3] (rewritten in the case α = 1, and taking into account all the simplifications implied by the assumptions ( 19)-( 23)), then carry out the same steps as in [ [24], §3] until Eq.(3.17) on page 409.This is sufficient because we can always assume R large enough and there is no need to remove the restriction contained in (3.17).This gives an analogue of Lemma 3.2 of [ [24], §3] valid in the present context.A weak Harnack inequality can be derived by an adaptation of the procedure introduced by Krylov and Safanov [ [23], §2] for the continuous case.Harnack principle follows then as a direct consequence of the weak Harnack inequality (see [24], §4).
In the proof of Theorem 1, together with Theorem 2 we need the following boundary variant of (24).In classical potential theory, the boundary Harnack principle describes the boundary behavior of positive harmonic functions vanishing on a portion of the boundary [1,21].It asserts that two positive harmonic functions vanishing on a portion of the boundary decay at the same rate.This principle generalizes to Π-caloric functions.The proof in [ [35], §5.1, §5.2 and §5.3], given for nonnegative L-caloric functions in cylindrical domains, is based only on the maximum and Harnack principles and can be readily extended to Π-caloric functions.Theorem 3. Let Γ and Π satisfying (19)- (23).Then, there exists a constant K > 1 such that for all s ∈ N, R > K and all couple of nonnegative Π-caloric functions We now have all the ingredients to prove Theorem 1.The kernel π defined by ( 8) satisfies all the conditions required in order to apply Theorems 2 and 3. Γ even , Γ odd satisfy in an obvious way (19).( 21) is implied by the normalization condition (6).( 22) is a consequence of ( 10) and ( 23) follows from (16).
Proof Theorem 1.Note that in the proof of Theorem 1 we can assume n C for a large constant C > 0. Otherwise (4) becomes evident because 1 ν n (x, y) 8 C , 1 C xy n 1 if we assume 1 x , y √ n and n C. Assuming n large enough, it becomes possible to apply Theorem 3. The crucial observation is that each of the functions (n; (x, y)) −→ u(x, y) and (n; (x, y)) −→ U(n; (x, y)), initially defined on N × Q, can be extended to all Z × Q.As for the first, this simply happens because it is independent of n; as for the second, the extension can be done by setting (25) U(n; (x, y)) ≡ 1, (n; (x, y)) ∈ (Z − ) × Q.
The second relation in the boundary conditions (2) satisfied by ν(n; (x, y)) guarantees that the extension ( 25) is π-caloric in Z × Q.The idea of such a construction is inspired by the proof of Lemma 4.1 in [38].
We now have at our disposal two positive π-caloric functions satisfying conditions of Theorem 3 that we will be able to compare.Setting v 1 ((x, y); n) = U(n; (x, y)), v 2 ((x, y); n) = u(x, y), s = n and R = √ n/4K, we deduce then that

For a cylindrical
subset B = A × {a n b} ⊂ Z 2 × Z where a < b ∈ Z we define the lateral boundary and the parabolic boundary of B by ∂ l B = a<n<b ∂ Γ A × {n}, ∂ p B = ∂ l B ∪ A × {a} , and we let B = B ∪ ∂ p B.
[32]ollows that the number of walks e n of length n taking steps in S, beginning and ending at the origin, and never leaving the positive orthant satisfiese n = O C|S| n n 3d/2 ,which gives an alternative proof of Theorem 7.2 of[32].