  Let $I$ be an independent set drawn from the discrete $d$-dimensional
hypercube $Q_d=\{0,1\}^d$ according to the hard-core distribution with
parameter $\lambda>0$ (that is, the distribution in which each independent set
$I$ is chosen with probability proportional to $\lambda^{|I|}$). We show a
sharp transition around $\lambda=1$ in the appearance of $I$: for $\lambda>1$,
$\min\{|I \cap {\cal E}|, |I \cap {\cal O}|\}=0$ asymptotically almost surely,
where ${\cal E}$ and ${\cal O}$ are the bipartition classes of $Q_d$, whereas
for $\lambda<1$, $\min\{|I \cap {\cal E}|, |I \cap {\cal O}|\}$ is
asymptotically almost surely exponential in $d$. The transition occurs in an
interval whose length is of order $1/d$.
  A key step in the proof is an estimation of $Z_\lambda(Q_d)$, the sum over
independent sets in $Q_d$ with each set $I$ given weight $\lambda^{|I|}$
(a.k.a. the hard-core partition function). We obtain the asymptotics of
$Z_\lambda(Q_d)$ for $\lambda>\sqrt{2}-1$, and nearly matching upper and lower
bounds for $\lambda \leq \sqrt{2}-1$, extending work of Korshunov and
Sapozhenko. These bounds allow us to read off some very specific information
about the structure of an independent set drawn according to the hard-core
distribution.
  We also derive a long-range influence result. For all fixed $\lambda>0$, if
$I$ is chosen from the independent sets of $Q_d$ according to the hard-core
distribution with parameter $\lambda$, conditioned on a particular $v \in {\cal
E}$ being in $I$, then the probability that another vertex $w$ is in $I$ is
$o(1)$ for $w \in {\cal O}$ but $\Omega(1)$ for $w \in {\cal E}$.
