Generalized monotonicity of transience for growing graphs

Establish whether transience is preserved when one deterministic non-decreasing sequence of uniformly bounded-degree graphs is contained edgewise at every time in another such sequence: specifically, prove or disprove that if the walk on (G_t) is transient, then the walk on (G'_t) is also transient whenever G_t ⊆ G'_t for all t.

Background

The paper proves recurrence and transience monotonicity for a single monotone increasing conductance environment relative to its initial or limiting static network. It leaves unresolved a stronger comparison principle involving two simultaneously growing graph sequences. Both sequences are required to have uniformly bounded degrees, and the larger graph sequence contains the smaller one at every time. The question asks whether transience of the walk in the smaller evolving graph forces transience in the larger evolving graph.

The authors note that this conjecture would imply the previously studied recurrence conjecture for growing graphs, although that recurrence result is separately established in the paper. The uniformly bounded-degree condition is essential: the paper gives an unbounded-conductance example showing that an analogous statement can fail without an appropriate boundedness assumption.

References

We were unable to establish this conjecture but find it very interesting, and restate it below. Let $(G_t){t\geq0}$ and $(G_t'){t\geq0}$ be two deterministic sequences of non-decreasing graphs of uniformly bounded degrees on the same countable vertex set $V$, with $G_t \subseteq G_t'$ for all $t$. Let $(X_t){t\geq0}$ and $(Y_t){t\geq0}$ be the corresponding walks on $(G_t){t\geq0}$ and $(G_t'){t\geq0}$, respectively, both starting at $v_0$. Then if $(X_t){t\geq0}$ is transient, in the sense that it almost surely returns to $v_0$ finitely often, then so is $(Y_t){t\geq0}$.

— Recurrence and transience of random walks on monotonically changing environments  (2609.28848 - Li et al., 23 Sep 2026) in Section 6, Section “Open questions,” Conjecture 1 (citing Conjecture 1.8 of Dembo, Huang, and Sidoravicius)