MERW transition probabilities for the growing pyramidal model
Determine whether the unique maximal-entropy random walk for the growing pyramidal model with bases of maximum size three has transition probabilities \[ p((x,y,z);(u,v,w))=\frac{H_{v-u,v-w}}{3H_{y-x,y-z}}, \] whenever the permissible configuration \((u,v,w)\) is obtained from \((x,y,z)\) by incrementing one coordinate, where \(H\) is the unique non-negative function on \(\mathbb Z^2\) satisfying \(H_{i,j}=0\) for \(i,j<0\), \(H_{0,0}=1\), and \(3H_{i,j}=H_{i-1,j}+H_{i,j-1}+H_{i,j}\) whenever \(i\geq 0\) or \(j\geq 0\).
References
Unfortunately, the generating functions in focus on $c_n(i,j)$, the number of walks of length $n$ starting from the origin and ending at $(i,j)$, which is not directly applicable here. Nonetheless, it is reasonable to conjecture that $a_n(i,j)$ adheres to the same asymptotic as $a_n(0,0)$, modulo a constant $H_{i,j}$. Consequently, $H$ can be identified as a unique positive harmonic function for Kreweras's random walk in the three-quarter plane, with $H_{0,0} = 1$, as established by . We make the following conjecture. The unique MERW for the growing pyramidal model depicted in Figure \ref{krew} is characterized by the following transition probabilities. Given $(u,v,w)$ as a permissible configuration derived from $(x,y,z)$ by incrementing one of its coordinates by $1$:
p((x,y,z);(u,v,w)) = \frac{H_{v-u,v-w}}{3 H_{y-x,y-z}},
where $H$ is the unique non-negative solution on $\mathbb Z2$ of $H_{i,j}=0$ for $i,j< 0$, $H_{0,0}=1$ and for all $i\geq 0$ or $j\geq 0$, $3 H_{i,j}=H_{i-1,j}+H_{i,j-1}+H_{i,j}$.