Phase Transition (A)=(4-1)a- 入 (△-2)A non-uniqueness regime uniqueness regime 2 4 6 8 10 △ sampling(with bounded error)from the hardcore model on graphs of max-degree A and fugacity A0Phase Transition 2 4 6 8 10 1 2 3 4 5 6 Δ λ non-uniqueness regime uniqueness regime ￾c(￾) = (￾ ￾ 1)(￾￾1) (￾ ￾ 2)￾ sampling (with bounded error) from the hardcore model on graphs of max-degree Δ and fugacity λ>0