Marginal Probability undirected graph G=(V,E) finite integer g≥2 Gibbs distribution uG over all configurations in [g] V possible boundary condition o E [g]A specified on an arbitrary subset ACD marginal distribution at vertexy: VxE [q]:u()=Pr [X=XA=0] XoHG a）=4o)）=Π-1(a) ZG i=1
Marginal Probability Gibbs distribution µG over all configurations in [q]V undirected graph G = (V, E) finite integer q ≥ 2 specified on an arbitrary subset Λ⊂V ∀ possible boundary condition σ ∈ [q]Λ marginal distribution at vertex µ v ∈ V : v 8x 2 [q] : µ v (x) = Pr X⇠µG [Xv = x | X⇤ = ] w() ZG = µ() = Y n i=1 µ1,...,i1 vi (i)
Marginal Probability undirected graph G=(V,E) finite integer q≥2 Gibbs distribution uG over all configurations in [g] V possible boundary condition o E [g]A specified on an arbitrary subset ACD marginal distribution g at vertex vEV: x∈[q]：%(x)=Pr[Xu=x|XA=o] XouG approximately approximately computingu computing ZG approx.inference approx.counting
Marginal Probability Gibbs distribution µG over all configurations in [q]V undirected graph G = (V, E) finite integer q ≥ 2 specified on an arbitrary subset Λ⊂V ∀ possible boundary condition σ ∈ [q]Λ marginal distribution at vertex µ v ∈ V : v 8x 2 [q] : µ v (x) = Pr X⇠µG [Xv = x | X⇤ = ] approximately computing µ v approximately computing ZG approx. inference approx. counting
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 tree
Spatial 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 infinitevolume 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()
Spatial Mixing (Decay of Correlation) marginal distribution at vertex v conditioning on weak spatial mixing（WsM)at rateδ()： ∀o,T∈[gA:lμg-lTv≤(distc(w,△) strong spatial mixing(SSM)at rateδ()： Vo,T∈[g]that differ on△：lμg-utTv≤(distc(v,△) SSM dist(v,△) marginal probabilities are well approximated by the local information
Spatial Mixing (Decay of Correlation) strong spatial mixing (SSM) at rate δ( ): weak spatial mixing (WSM) at rate δ( ): µ v : marginal distribution at vertex v conditioning on σ SSM marginal probabilities are well approximated by the local information G v dist(v,Δ) Δ Λ\Δ weak spatial mixing (WSM) at rate δ( ): 8, ⌧ 2 [q] ⇤ : kµ v µ⌧ v kT V (distG(v,⇤)) kµ v µ⌧ 8, ⌧ 2 [q] v kT V (distG(v, )) ⇤ that differ on Δ:
Tree Recurrence hardcore model:i independent set I in T u( pr≌，Pr[is unoccupied by I] IouT T Wd T T Ta pz,≌，Pr[is unoccupied byI] IouTi
pT , Pr I⇠µT [v is unoccupied by I ] Ti v ui T hardcore model: independent set I in T µT (I) / |I| Tree Recurrence pTi , Pr I⇠µTi [ui is unoccupied by I ] u1 ud T1 Td