Abstract
In this article we are interested in finding positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane—resolution of a functional equation via boundary value problem using a conformal mapping—to the three-quarter plane applying the strategy of splitting the domain into two symmetric convex cones. We obtain a simple explicit expression for the algebraic generating function of harmonic functions associated to random walks avoiding a quadrant.
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Acknowledgments
I would like to thank Kilian Raschel for his strong support, advice and suggestions. I also would like to thank Samuel Simon for his help with the English. Finally, I thank the anonymous referee for his/her comments and suggestions. Most of the work presented in this article was done during my affiliation at Department of Mathematics, Simon Fraser University, Canada & Institut Denis Poisson, Université de Tours et Université d’Orléans, France.
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A. Trotignon was supported by the Austrian Science Fund (FWF) grant FWF05004.
Appendices
Appendix A: Non-symmetric Case
In this section (except clearly stated) we suppose that the probability transitions of the random walks we consider are not symmetric and a priori neither is the harmonic function. In other words, the random walks satisfy hypotheses (H1), (Ĥ2), (H3) and (H4) with
-
(Ĥ2)
We assume that the transition probabilities p0,0 = p− 1,1 = p1,− 1 = 0.
and discrete harmonic functions associated to these random walks satisfy the properties (P1), (P2), (P3) but not necessarily (P4).
Like in Section 2.3, we split the three-quadrant into two symmetric convex cones of opening angle \(\frac {3\pi }{4}\), and split the generating function H(x, y) into three generating functions: L(x, y) for harmonic function in the lower part, D(x, y) for the ones on the diagonal and U(x, y) for the ones in the upper part (see Eq. 19). Since we do not have the symmetry conditions anymore, instead of one functional equation as in Lemma 4, we end up with a system of two functional equations.
Lemma 20
For any random walks with property (H1), the generating functions L(x, y) and U(x, y) satisfy the following system of functional equations
Remark 21
Notice the symmetry between x and y in Eq. 58. In the case of a symmetric random walk (i.e. if pi, j = pj, i) the two functional equations are the same.
Proof
The strategy of the proof is similar to the proof of Lemma 4. In addition to the functional equations for L(x, y) and D(x, y), see Eqs. 21 and 22, we can write a functional equation for U(x, y):
Mixing (21), (59) and (58) and multiplying by xy, we get (1). □
In this system of two functional equations, the bivariate generating function U(x, y) (resp. L(x, y)) is related to the bivariate generating functions D(x, y), Dℓ(x, y) (resp. Du(x, y)) and the univariate generating function U− 0(x− 1) (resp. L0−(y− 1)). In order to simplify this system of functional equations, we perform two changes of variables φL for the lower part and φU for the upper part with
This change of variables transforms the upper part {j ≥ 1,i ≤ j − 1} into the left quadrant {i ≤− 1,j ≥ 0} and the lower part {i ≥ 1,j ≤ i − 1} into the right quadrant {i ≥ 1,j ≥ 0} (see Fig. 9). Note that the diagonal is changed by both φL and φU into the positive y-axis (see Fig. 9). The step set is changed as well, and walks in the three-quadrant can be seen as inhomogeneous walks in the half-plane with two different step sets in each quadrant and a mixed step set on the positive y-axis (see Fig. 10). Other type of inhomogeneous walks are studied in [5, 8].
In the symmetric case (symmetry of the random walks and hence the symmetry of the unique positive harmonic function), the study of harmonic functions of random walks in the three quadrant is then equivalent to the study of harmonic functions of random walks reflected on the y-axis and constrained by the x-axis in the positive quadrant (see Fig. 11). Articles [2, 21] work on problem in the same vein: their authors study walks in the quadrant with different weights on the boundary.
Unfortunately, due to number of unknown functions, we are not able to solve this system of functional equations yet. Let us end this section by pointing out that the split cone along the diagonal can also be related to the Join-the-Shortest-Queue model (JSQ). Supposed that there are two lines (see Fig. 12, left), each of them with a service time exponentially distributed of rate r1 and r2 and that the customers arrive according to a Poisson process. The clients choose the shorter queue and if both lines happened to have the same length, the costumers pick one or the other with probability p1 or p2. A common question in queuing theory is to obtain closed-form expression for the stationary distribution. This JSQ problem can be modeled by random walks in the quarter plane split into two octants, each axis representing the length of each line (see Fig. 12, right). The symmetric case (when r1 = r2 and p1 = p2 = 1/2) is solvable, see [1, 15, 16] and [13, Chap. 10] for reference. On the other side, the non-symmetric case is still an open problem.
Appendix B: Discrete Harmonic Functions in the Quadrant
Even if discrete harmonic functions of random walks in the quarter plane are already studied in [19], we want to point out that the strategy of splitting the domain into three parts (the upper part, the diagonal and the lower part) can also be performed in the quarter plane \(\mathcal Q\) defined in Eq. 15 and allow us to find explicit expressions of harmonic functions. We suppose that random walks satisfy the hypotheses (H1), (H2), (H3), (H4), and associated discrete harmonic functions \(\widetilde {f}\) the properties (P1), (P2), (P3). The generating function of such harmonic functions is defined in Eq. 16. We split the quadrant \(\mathcal Q\) in three parts: the lower part {i ≥ 2, 1 ≤ j ≤ i − 1}, the diagonal and the upper part {j ≥ 2, 1 ≤ i ≤ j − 1}. As in Eq. 19, by construction we have
where \(\widetilde {L}(x,y)=\underset {\underset {1\leq j\leq i-1}{i \geq 2}}{\sum }\widetilde {f}(i,j)x^{i-1} y^{j-1}\) denotes the generating function of harmonic functions associated to random walks ending in the lower part, \(\widetilde {D}(x,y)=\underset {i\geq 1}{\sum }\widetilde {f}(i,i)x^{i-1} y^{i-1} \) ending on the diagonal and \(\widetilde {U}(x,y)=\underset {\underset {1\leq i\leq j-1}{j \geq 2}}{\sum }\widetilde {f}(i,j)x^{i-1} y^{j-1}\) in the upper part (see Fig. 13).
We can write a functional equation for each section and finally get a functional equation in terms of \(\widetilde {L}(x,y)\), \(\widetilde {L}_{-0}\) and \(\widetilde {D}(x,y)\).
Lemma 22
For any random walks with property (H1) and (H2), the generating function \(\widetilde {L}(x,y)\) satisfies the following functional equation
with \(\widetilde {L}_{-0}(x)= {\sum }_{i\geq 2} \widetilde {f}(i,1)x^{i-1}\).
In order to simplify the last functional equation (61), we apply the following change of variables
The Eq. 61 is changed into
with
and
The angle 𝜃ψ is also simply related to 𝜃 (see Fig. 2b) as
The scheme of the proof is the same as the proof of Lemma 10. Writing \( \theta _{\psi }= \arccos (-c_{\psi })\), we have \(c_{\psi }=-\sqrt {\frac {1-c}{2}}\), hence Eq. 66. The functional (63) is very close to the functional equation in the three-quarter plane (25), the only difference is the additional term p0,1x. We can then write a similar boundary value problem as Lemma 11.
Lemma 23
The generating function \(\widetilde {D}_{\psi }(y)\) is analytic in \(\mathcal {G}_{\mathcal L_{\psi }}\) and continuous on \(\overline {\mathcal {G}_{\mathcal L_{\psi }}}\setminus \{1\}\). Moreover, for all \(y\in \mathcal L_{\psi }\setminus \{1\}\), \(\widetilde {D}_{\psi }(y)\) satisfies the following boundary condition
where \(\frac {g_{\psi }(y)}{g_{\psi }^{\prime }(y)}=\frac {\widetilde {w}_{\psi }(y)\sqrt {\widetilde {w}_{\psi }(Y_{\varphi }(x_{\psi ,1}))-\widetilde {w}_{\psi }(y)}}{\widetilde {w}_{\psi }^{\prime }(y)\sqrt {\widetilde {w}_{\psi }(Y_{\psi }(x_{\psi ,1}))}}\) and \(\widetilde {w}_{\psi }\) is a conformal gluing function defined in Section 3.1.
Moreover, the expression of \(\widetilde {D}_{\psi }(y)\) is the same as in the three-quarter plane and we have the following theorem, which is a quarter plane equivalent to Theorem 13.
Theorem 24
The generating function Dψ(y) can be written as
with 𝜃ψ, \(\widetilde {w}_{\psi }(y)\) and \(\widetilde {W}_{\psi }\) defined in Section 3.1 and \(\widetilde {\delta }_{\psi }\) in Eq. 65.
From the expression of \(\widetilde {D}_{\psi }\) in Eq. 68 and the functional (63), we get an expression for \(\widetilde {L}_{\psi }(x,0)\):
and from Eq. 30 there exists k≠ 0 such that, for x in the neighborhood of 1,
This asymptotic results matches that in [19]. We end this section with the example of the simple random walk. The application ψ changes the simple random walk into the Gouyou-Beauchamps random walk (see Fig. 14).
In order to compute \(\widetilde {L}(x,0)\), we need to calculate 𝜃ψ, \(\widetilde {\delta }_{\psi }\), \(\widetilde {w}_{\psi }\) and Wψ. We easily have \(\theta _{\psi }=\frac {\pi }{4}\) and
On the other side, we have
Finally, noticing that \(K(x,0)=\frac {x}{4}\) and that \(\widetilde {H}(x,0)=\widetilde {f}(1,1)+\widetilde {L}(x,0)\), from Eq. 69 we get
and this result matches the computation in [19, Eq. 2.6].
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Trotignon, A. Discrete Harmonic Functions in the Three-Quarter Plane. Potential Anal 56, 267–296 (2022). https://doi.org/10.1007/s11118-020-09884-y
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DOI: https://doi.org/10.1007/s11118-020-09884-y