Spatial Mixing (Decay of Correlation) marginal distribution at vertex v conditioning on weak spatial mixing（WsM)）at rateδ()： o,T∈[gl：‖lg-IlTv≤6(dista(v,A) boundary conditions 0 dist(v,A) on infinite graphs: WSM 1<> uniqueness of infinite- volume Gibbs measure 入≤λ(A)<> WSM of hardcore model on infinite A-regular treeSpatial Mixing (Decay of Correlation) G v dist(v,Λ) weak spatial mixing (WSM) at rate δ( ): µ￾ v : marginal distribution at vertex v conditioning on σ boundary conditions on infinite graphs: WSM uniqueness of infinite￾volume Gibbs measure ￾ µ￾ v WSM of hardcore model on infinite Δ-regular tree Λ 8￾, ⌧ 2 [q] ⇤ : kµ￾ v ￾ µ⌧ v kT V  ￾(distG(v,⇤)) ￾  ￾c(￾)