What kinds of graphs are known to exhibit sharp threshold for independent bernoulli percolation? Here, sharp threshold stands for exponential decay of the probability of the cluster range below the critical point $p_c$ and superlinear growth of $\theta$ above $p_c$ (the mean field lower bound) .
More precisely, i would like to know for what classes of graphs $G$ the following two properties (which are well known for $\mathbb{Z}^d$) hold. If $B_n$ is a ball of radius $n$ centered at the origin (denoted by $0$), $$ \mathbb{P}_p(0 \text{ is connected to }\partial B_n)\leq e^{-cn}, \text{ for }p<p_c. $$ and $$ \theta(p):=\mathbb{P}_p(0\leftrightarrow \infty)\geq c(p-p_c), \text{ for }p>p_c. $$
sharp thresholds for percolation in a more general meaning are also welcome.
An interesting proof of the statement you make is with Hugo Duminil-Copin and Vincent Tassion, in this article: https://arxiv.org/pdf/1502.03050.pdf