Uniqueness Threshold Pr[v∈I|o] (I)x入I △-regular tree →∞ 0-O○ o:boundary condition fixing each leaf EI org Critical phenomenon: Pr[v∈Ilo]is independent of o when /→∞ f入≤入c(△)= (△-1)A-1) (△-2)AUniqueness Threshold regular tree ` ! 1 v σ: boundary condition fixing each leaf ∈I or ∉I Pr[v 2 I | ￾] Critical phenomenon: random independent set I iff Δ- ￾  ￾c(￾) = (￾ ￾ 1)(￾￾1) (￾ ￾ 2)￾ Pr[v∈I | σ] is independent of σ when l→∞