Using potential theoretic techniques, we show how it is possible to determine the dominant asymptotics for the number of walks of length $n$, restricted to the positive quadrant and taking unit steps in a balanced set $\Gamma$. The approach is illustrated through an example of inhomogeneous space walk. This walk takes its steps in $\{ \leftarrow, \uparrow, \rightarrow, \downarrow \}$ or $\{ \swarrow, \leftarrow, \nwarrow, \uparrow,\nearrow, \rightarrow, \searrow, \downarrow \}$, depending on the parity of the coordinates of its positions. The exponential growth of our model is $(4\phi)^n$, where $\phi= \frac{1+\sqrt 5}{2}$denotes the Golden ratio, while the subexponential growth is like $1/n$.As an application of our approach we prove the non-D-finiteness in two dimensions of the length generating functions corresponding to nonsingular small step sets with an infinite group and zero-drift.
@article{10_37236_5877,
author = {Philippe D'Arco and Valentina Lacivita and Sami Mustapha},
title = {Combinatorics meets potential theory},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {2},
doi = {10.37236/5877},
zbl = {1336.05011},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5877/}
}
TY - JOUR
AU - Philippe D'Arco
AU - Valentina Lacivita
AU - Sami Mustapha
TI - Combinatorics meets potential theory
JO - The electronic journal of combinatorics
PY - 2016
VL - 23
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/5877/
DO - 10.37236/5877
ID - 10_37236_5877
ER -
%0 Journal Article
%A Philippe D'Arco
%A Valentina Lacivita
%A Sami Mustapha
%T Combinatorics meets potential theory
%J The electronic journal of combinatorics
%D 2016
%V 23
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/5877/
%R 10.37236/5877
%F 10_37236_5877
Philippe D'Arco; Valentina Lacivita; Sami Mustapha. Combinatorics meets potential theory. The electronic journal of combinatorics, Tome 23 (2016) no. 2. doi: 10.37236/5877
