Quantitative Benjamini conjecture for heat-kernel expansion

Let $\mathcal{G}$ be an infinite, irreducible, bounded-degree graph, let $\nu_{L,o}$ denote the $L$-step heat-kernel measure rooted at $o$, and let $\partial A$ be the boundary of $A\subset\mathcal{G}$. Quantitative Benjamini conjecture. There exist a set $A\subset\mathcal{G}$ and a vertex $o$ such that

$$\nu_{L,o}(A)<1/2$$

and

$$\nu_{L,o}(\partial A)\geq \phi^L\nu_{L,o}(A)$$

for $\phi<1$. This is proposed as a quantitative refinement of Benjamini's conjecture and would yield initial states with relaxation times exponential in system size; the paper discusses related results and proves the refinement for dynamics arising from hyperbolic groups.