Phase transition for the extended-action square-lattice process

Determine whether the phase transition of the two-dimensional PMDP with admissible steps \(\{(0,1),(1,0),(2,0)\}\) coincides with the phase transition of the corresponding oriented percolation process on the square lattice.

Background

The paper compares the oriented-percolation PMDP with another two-dimensional example whose action set includes a dependent action, corresponding to admissible steps (0,1)(0,1), (1,0)(1,0), and (2,0)(2,0). Numerical simulations suggest that its limit value also exhibits a transition to value 1.

The authors do not establish the identity of the associated percolation process or prove that the two phase-transition parameters coincide; they explicitly formulate this as a conjecture.

References

It appears that Example~\ref{example:extended_moves}, like the oriented-percolation example, exhibits a phase transition at which the limit value becomes equal to $1$. We expect this transition to coincide with the phase transition of a corresponding percolation process. We conjecture that this process is the one on the square lattice whose admissible paths have step set ${(0, 1), (1, 0), (2, 0)}$.

— Percolation Markov Decision Processes  (2609.19905 - García et al., 17 Sep 2026) in Section 7.4, caption and discussion surrounding Figure 2

For oriented PMDPs with $\mathrm{Bernoulli}(p)$ payoffs and $p \in (0, 1)$, the uniform value, as a function of $p$, undergoes a phase transition at the same critical parameter as an associated oriented percolation process defined on the same transition graph.

— Percolation Markov Decision Processes  (2609.19905 - García et al., 17 Sep 2026) in Section 8, Perspectives, Point 4