  We establish tight results for rapid mixing of Gibbs samplers for the
Ferromagnetic Ising model on general graphs. We show that if
\[(d-1)\tanh\beta<1,\] then there exists a constant C such that the discrete
time mixing time of Gibbs samplers for the ferromagnetic Ising model on any
graph of n vertices and maximal degree d, where all interactions are bounded by
$\beta$, and arbitrary external fields are bounded by $Cn\log n$. Moreover, the
spectral gap is uniformly bounded away from 0 for all such graphs, as well as
for infinite graphs of maximal degree d. We further show that when
$d\tanh\beta<1$, with high probability over the Erdos-Renyi random graph
$G(n,d/n)$, it holds that the mixing time of Gibbs samplers is
\[n^{1+\Theta({1}/{\log\log n})}.\] Both results are tight, as it is known that
the mixing time for random regular and Erdos-Renyi random graphs is, with high
probability, exponential in n when $(d-1)\tanh\beta>1$, and $d\tanh\beta>1$,
respectively. To our knowledge our results give the first tight sufficient
conditions for rapid mixing of spin systems on general graphs. Moreover, our
results are the first rigorous results establishing exact thresholds for
dynamics on random graphs in terms of spatial thresholds on trees.
