  We prove that if $(G_n)_{n\geq1}=((V_n,E_n))_{n\geq 1}$ is a sequence of
finite, vertex-transitive graphs with bounded degrees and $|V_n|\to\infty$ that
is at least $(1+\epsilon)$-dimensional for some $\epsilon>0$ in the sense that
\[\mathrm{diam} (G_n)=O\left(|V_n|^{1/(1+\epsilon)}\right) \text{ as
$n\to\infty$}\] then this sequence of graphs has a non-trivial phase transition
for Bernoulli bond percolation. More precisely, we prove under these conditions
that for each $0<\alpha <1$ there exists $p_c(\alpha)<1$ such that for each
$p\geq p_c(\alpha)$, Bernoulli-$p$ bond percolation on $G_n$ has a cluster of
size at least $\alpha |V_n|$ with probability tending to $1$ as $n\to \infty$.
In fact, we prove more generally that there exists a universal constant $a$
such that the same conclusion holds whenever \[\mathrm{diam}
(G_n)=O\left(\frac{|V_n|}{(\log |V_n|)^a}\right) \text{ as $n\to\infty$.}\]
This verifies a conjecture of Benjamini up to the value of the constant $a$,
which he suggested should be $1$. We also prove a generalization of this result
to quasitransitive graph sequences with a bounded number of vertex orbits and
prove that one may indeed take $a=1$ when the graphs $G_n$ are all Cayley
graphs of Abelian groups.
  A key step in our proof is to adapt the methods of Duminil-Copin, Goswami,
Raoufi, Severo, and Yadin from infinite graphs to finite graphs. This
adaptation also leads to an isoperimetric criterion for infinite graphs to have
a nontrivial uniqueness phase (i.e., to have $p_u<1$) which is of independent
interest. We also prove that the set of possible values of the critical
probability of an infinite quasitransitive graph has a gap at $1$ in the sense
that for every $k,n<\infty$ there exists $\epsilon>0$ such that every infinite
graph $G$ of degree at most $k$ whose vertex set has at most $n$ orbits under
Aut$(G)$ either has $p_c=1$ or $p_c\leq 1-\epsilon$.
