\lambda_c(\Delta)$, the problem of sampling independent sets with $\Pr[I] \propto \exp(\lambda|I|)$ on $\Delta$-regular graphs has a fully polynomial time approximation scheme.

NP-hard for large $\lambda$:

:   For any $\lambda >\lambda_c(\Delta)$, unless $NP=RP$, there is no fully polynomial time randomized approximation scheme for the problem of sampling independent sets with $\Pr[I] \propto \exp(\lambda|I|)$ on $\Delta$-regular graphs.

There is a family ${\ensuremath{\mathcal{F}}}$ of functions $f : {B_d(0, R)}\rightarrow \Re$ such that the following hold.

Efficiency: 

:   Every $f \in {\ensuremath{\mathcal{F}}}$ is computable in time $poly(d)$.

Sampling has a threshold:

:   There is a constant $\lambda_c > 0$ such that for any $\frac 1 d < \lambda < \lambda_c$, there is a $poly(d/\eta)$-time $\eta$-sampler from from the distribution ${{\mathcal{D}}_f^{\lambda}}$. On the other hand, for $\lambda > \lambda_c$, there is a constant $\eta' >0$ such that no polynomial time algorithm $\eta'$-samples from ${{\mathcal{D}}_f^{\lambda}}$ unless $NP=RP$.

For a graph $G = (V, E)$, the function $h$ on the hypercube ${\mathbb{H}_{d}}$ is defined in the natural way for $d=|V|$. We identify $\vy$ with $V_{\vy} = \{v_i : \vy_i = 1\}$, and set $h(\vy)=d - |V_\vy|$ if $V_{\vy}$ is an independent set in $G$ and to $d$ otherwise. We then apply Theorem \[thm:discrete\_to\_cont\_v1\], with $R=2\sqrt{d}$ and $M 