Spatial Mixing (Decay of Correlation) marginal distribution at vertex v conditioning on weak spatial mixing(WSM）at rateδ()： o,T∈[g∂R：l呀-呀ITv≤δ(t) boundary conditions R 0 on infinite graphs: WSM uniqueness of infinite- volume Gibbs measure λ≤入c= dd a-a可> WSM of hardcore model on infinite(d+1)-regular treeSpatial Mixing (Decay of Correlation) R G v t weak spatial mixing (WSM) at rate δ( ): kµ￾ v ￾ µ⌧ 8￾, ⌧ 2 [q] v kT V  ￾(t) @R : µ￾ v : marginal distribution at vertex v conditioning on σ boundary conditions on infinite graphs: WSM uniqueness of infinite￾volume Gibbs measure ￾ µ￾ v ￾  ￾c = dd (d￾1)(d+1) WSM of hardcore model on infinite (d+1)-regular tree ∂R