Constant-factor repair of decreasing transience monotonicity

Determine whether a monotone non-increasing conductance environment remains transient when its limiting conductances satisfy $c_\infty\geq\alpha c_0$ for some constant $\alpha>0$ and the limiting static network is irreducible and transient.

Background

The paper disproves the conjecture that transience of the limiting network implies transience of a monotone decreasing dynamic walk. Its counterexample allows conductances to be removed entirely, so the limiting conductances are not bounded below by a fixed positive multiple of the initial conductances.

The authors explicitly ask whether the failed conjecture can be restored under the constant-factor condition c∞≥αc0c_\infty\geq\alpha c_0. This condition is analogous to the assumption imposed in the separate decreasing-recurrence conjecture.

References

While both \cref{conj:dec_trans,conj:dec_rec} are false, we do not know whether \cref{conj:dec_trans} can be repaired or not by assuming such a constant-factor assumption. Suppose random walk $(X_t){t\geq0}$ in changing environment corresponding to deterministic conductances $(c_t){t\geq0}$ has $(c_t){t\geq0}$ non-increasing with limit $c\infty$, i.e., $c_t\downarrow c_\infty$. If $c_\infty\geq\alpha c_0$ for some $\alpha>0$ and random walk corresponding to $c_\infty$ is irreducible and transient, must $(X_t)_{t\geq0}$ also be transient, in the sense that almost surely the walk visits every vertex finitely often?

— Recurrence and transience of random walks on monotonically changing environments  (2609.28848 - Li et al., 23 Sep 2026) in Section “Open questions,” Question 1