Abstract
An upper bound and finiteness criteria for the Galois group of weighted walks with rational coefficients in the quarter plane
Highlights
Counting lattice walks is a classic problem in combinatorics
If a multiple step length requirement is allowed, a combinatorial walk can be seen as a weighted walk with integer weights
If we allow the weights of a walk to take arbitrary non-negative real values that sum to 1, we arrive at the realm of probabilistic walks in the quarter plane
Summary
A combinatorial walk with nearestneighbour step length can be seen as a weighted walk with weight 1 for the allowed directions and weight 0 for the forbidden directions. If a multiple step length requirement is allowed, a combinatorial walk can be seen as a weighted walk with integer weights. For a weighted walk, we may assume that the weights sum to 1 by normalization. If we allow the weights of a walk to take arbitrary non-negative real values that sum to 1, we arrive at the realm of probabilistic walks in the quarter plane. A weighted walk is the same thing as a probabilistic walk and a weighted walk with rational weights is the same thing as a combinatorial walk with different step lengths in different directions
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