Lattice walks in the octant with infinite associated groups

Abstract

Continuing earlier investigations of restricted lattice walks in N 3 , we take a closer look at the models with infinite associated groups. We find that up to isomorphism, only 12 different infinite groups appear, and we establish a connection between the group of a model and the model being Hadamard.

3D positive lattice walks and spherical triangles

“…To summarize, classifying the triangles is close to, but different from classifying the groups. The latter task has already been achieved in [2] (finite group case; we have reproduced their results in Table 4) and [49] (infinite groups; see our Table 1), using a heavy computer machinery. However, the group classification is more precise, in the sense that the spherical triangle does not determine everything: infinite group models can have a tiling triangle, and Hadamard models are not the only ones to have birectangular triangles.…”

“…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.…”

“…Section 5: We classify the models with respect to their triangle and the associated principal eigenvalue, and compare our results with the classification in terms of the group and the Hadamard property obtained in [13,49]. Finite group models correspond to triangular tilings of the sphere S 2 .…”

“…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.…”

“…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…”