Uniqueness Threshold Pr[w∈I|o] random independent setI (I)x入I △-regular tree o:boundary condition fixing each leaf El or g Critical phenomenon: Pr[v∈IlO]is independent of o when l-∞ f入≤A(△)= (△-1)A-1) e 义 (△-2)A △- 2 Uniqueness thresholdUniqueness 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)￾ ⇡ e ￾ ￾ 2 Pr[v∈I | σ] is independent of σ when l→∞ Uniqueness threshold