Uniformly bounded-degree counterexample for decreasing recurrence

Determine whether there exists a counterexample to the decreasing-environment recurrence conjecture in which the evolving networks have uniformly bounded degree or uniformly bounded total conductance at every vertex.

Background

The paper constructs a monotone non-increasing environment in which every static network in the evolution, including the limiting network, is recurrent, while the changing-environment walk is transient with probability at least 1−δ. The construction uses a base graph whose vertices have infinite degree and pendant rooms with conductances M_n that tend to infinity.

The authors explain that the construction can be modified to be locally finite, but they do not know whether a counterexample can satisfy the stronger uniform bounded-degree or uniform bounded-total-conductance requirements.

References

It is unclear to us whether a counterexample with uniformly bounded degree or uniformly bounded total conductance, i.e., $\pi_t(v) \leq M$ for all $t$ and $v$, exists; note that in our case, $M_n\to\infty$.

— Recurrence and transience of random walks on monotonically changing environments  (2609.28848 - Li et al., 23 Sep 2026) in Remark following Lemma 5.1, Section “A counterexample for decreasing recurrence”

Lastly, we remark that our counterexample for \cref{conj:dec_rec} in \cref{thm:dec_rec} is not as strong as it could potentially be, and we leave open the question of whether or not a stronger counterexample can be found. In particular, our counterexample only shows $(X_t){t\geq0}$ is transient with probability at least $1-\delta$, and one could hope to find a counterexample where $(X_t){t\geq0}$ is almost surely transient.

— Recurrence and transience of random walks on monotonically changing environments  (2609.28848 - Li et al., 23 Sep 2026) in Section “Open questions,” final paragraph before the acknowledgments