Infinite Orders and Non-D-finite Property of 3-Dimensional Lattice Walks

Abstract

Recently, Bostan and his coauthors investigated lattice walks restricted to the non-negative octant N 3 . For the 35548 non-trivial models with at most six steps, they found that many models associated to a group of order at least 200 and conjectured these groups were in fact infinite groups. In this paper, we first confirm these conjectures and then consider the non-D-finite property of the generating function for some of these models.

3D positive lattice walks and spherical triangles

“…• First of all, Corollary 7 is (to the best of our knowledge) the first non-D-finiteness result on truly 3D models (the 3D models considered in [32] have dimensionality 2 in the sense of Definition 1, and thus do not satisfy the main hypothesis (H), which guarantees the existence of a non-degenerate spherical triangle, see Section 7.5).…”

“…A model labeled "both" is simultaneously (1, 2)-type and (2, 1)-type Hadamard. The total number of models is computed in [13], the number of (in)finite groups in [13,32,49] and the refined statistics on 3D Hadamard models in [48] For each type, an example is presented in Figure 4: for the (2, 1)-type we have taken U (x, y) = x + x + y + y (the 2D simple walk, see Figure 7), V (x, y) = x + xy + xy + xy + y (a scarecrow model, see again Figure 7) and T (z) = z + z. For the (1, 2)-type we have χ(x, y, z) = U (z) + V (z)T (x, y) (permutation of the variables in the definition (9)), with U (z) = z + z, V (z) = z + 1 + z and T (x, y) the generating function of the same scarecrow model as above.…”

“…An algorithmic proof of the irrationality of such quantities is proposed in [18,Sec. 2.4], and further applied to some weighted models in [32].…”

“…In majority, the techniques used in [13] to solve finite group models are the algebraic kernel method and computer algebra (using the guessingand-proving paradigm). The classification (in particular with respect to the finiteness of the group and the Hadamard structure) of the 3D small step models with arbitrary cardinality is pursued in the articles [2,32,61,49]. Table 2 reproduces this classification.…”

“…Concerning the algebraic nature of the 3D generating function (1), a few results are known: in the finite group cases solved in [13], the generating function is always D-finite. On the other hand, the article [32] proves that for some degenerate (in the sense of the dimensionality) 3D models, the excursion generating function O(0, 0, 0; t) is non-D-finite, by looking at the asymptotic behavior of the excursion sequence and showing that λ in (2) is irrational, extending the work [18]. Does there exist a non-degenerateÂă3D finite group model with a non-D-finite generating function (1)?…”