Combinatorics Meets Potential Theory
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$ T he approach is illustrated through an example of inhomogeneous space walk This walk takes its st eps 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
Electron. J. Comb.