3D positive lattice walks and spherical triangles

in collaboration with Vincent Perrollaz, Kilian Raschel and Amelie Trotignon

For details and complete description of this work, take a look at our paper 3D positive lattice walks and spherical triangles.

  1. Introduction
  2. Presentation of the results
  3. Numerical tools


We consider 3D random walk models with steps in a family $\mathcal S \subset \{(x,y,z): x,y,z \in \{-1,0,1\}\}\setminus\{0,0,0\}$. For non degenerate models where $\mathcal S$ is not included in a half-space the critical exponent of each model can be linked to the fundamental eigenvalue of a spherical triangle.

Our results deal with the following aspects:

Presentation of the results

Theorem 17. (from our article) Among all infinite group 3D models we have the following classification in terms of angles:

Partitions associated to the equilateral triangles with angles $\pi/2$ and $3\pi/2$

Particular examples

Numerical tools

If you use any of the tools provided in this section in your work, please give credid by citing the associated article.

In this section we present some numerical tools used in our work. These tools can confirm some of the results presented in the paper.

  1. Computing the eigenvalue of a spherical triangle coupled with an extrapolation procedure in order to accelerate the convergence.
  2. Testing if a model is included in a halfspace
  3. Compute the associated triangle to a 3D walk model
  4. Compute critical exponent given a 3D model: Matlab implementation with numerical computation of the triangle and the eigenvalue: readDB_inf_mat.m
  5. Testing if the combinatorial group is a subgroup of the group of the symmetries of the triangle (with machine precision)
  6. Some numerical results (containing matrices which can be directly imported into Matlab)

Created: Mar 2018, Last modified: Mar 2018