How fast does the range of simple random walk grow?

Consider a discrete-time simple random walk $(X_t)_{t\ge 0}$ on an infinite, connected, locally finite graph $G$. Let $R_t := |\{X_0,\dots,X_t\}|$ denote its range at time $t$, and $T_n:=\inf\{t\ge 0: R_t= n\}$ the $n-$th discovery time. We establish a general estimate on $\mathbb E[T_n]$ in terms of two coarse geometric parameters of $G$, and deduce the universal bounds $\mathbb E[T_n]\le 4n^3\log n$ and $\mathbb E[R_t]\gtrsim (t/\log t)^{1/3}$. Moreover, we show that this is essentially sharp by constructing a multi-scale version of Feige's Lollipop graph satisfying $\mathbb E[T_n]\gtrsim n^{3}$ for all dyadic integers $n$. In light of this example, we ask whether the existence of \emph{trapping phases} where the range grows sub-diffusively necessarily implies the existence of \emph{expanding phases} where it grows super-diffusively. Finally, we provide a simple \emph{uniform transience} condition under which the expected range grows linearly, and conjecture that all vertex-nonamenable graphs exhibit linear range.