pr A Pr lv is unoccupied by I] IouT Zr(v is unoccupied) Zr(v is unoccupied)+Zr(v is occupied) T where Zr(event A)≌∑w(I) T I:A holds
pT , Pr I⇠µT [v is unoccupied by I ] Ti v ui T = ZT (v is unoccupied) ZT (v is unoccupied) + ZT (v is occupied) ZT (event A) , X I: A holds w(I) where
pr A Pr lv is unoccupied by I] IouT Zr(v is unoccupied) Zr(v is unoccupied)+Zr(v is occupied) T Product rule: d Zr(v is unoccupied)=Zr. i=1 T d Zr(v is occupied)=Zr(ui is unoccupied) i=1 Π12m I=1Zr+λΠ1Zm,(u,is unoccupied) 1 1+入Π1pm
pT , Pr I⇠µT [v is unoccupied by I ] Ti v ui T = ZT (v is unoccupied) ZT (v is unoccupied) + ZT (v is occupied) ZT (v is unoccupied) = Y d i=1 ZTi Product rule: = 1 1 + Qd i=1 pTi ZT (v is occupied) = Y d i=1 ZTi (ui is unoccupied) = Qd i=1 ZTi Qd I=1 ZTi + Qd i=1 ZTi (ui is unoccupied)
p7≌，Pr[is unoccupied by I|o] Zr(v is unoccupied∧o) Zr(v is unoccupied∧g)+Zr(v is occupied∧o) T Product rule: d Zr(v is unoccupied Ao)=Zr:(: 2=1 T Oi Zr(is occupied∧o)=入ΠZn.(u:is unoccupied∧a) i=1 Π1Zn,(o) Π=1Zr,+λI1Zm,(o)(u,is unoccupied∧o) 1 1+Π1%
Ti v ui T Product rule: i , Pr I⇠µT p [v is unoccupied by I | ] T = ZT (v is unoccupied ^ ) ZT (v is unoccupied ^ ) + ZT (v is occupied ^ ) ZT (v is unoccupied ^ ) = Y d i=1 ZTi (i) = 1 1 + Qd i=1 pi Ti ZT (v is occupied ^ ) = Y d i=1 ZTi (ui is unoccupied ^ i) = Qd i=1 ZTi (i) Qd I=1 ZTi + Qd i=1 ZTi (i)(ui is unoccupied ^ i)
Tree Recurrence hardcore model: independent set I in T uT( p≌，Pr[v is unoccupied by I|o] IouT T Occupancy ratio: Rg Prrv is occupied o] Prr[v is unoccupied o] T =(1-p%)/p d 1 p= 1 1+Π=1 =λΠ i=1 %+1
Ti v ui T hardcore model: independent set I in T µT (I) / |I| Tree Recurrence i , Pr I⇠µT p [v is unoccupied by I | ] T = 1 1 + Qd i=1 pi Ti p T Occupancy ratio: R T , PrT [v is occupied | ] PrT [v is unoccupied | ] = (1 p T )/p T R T = Y d i=1 1 Ri Ti + 1
Tree Recurrence 2-spin ystem=[ma6-卧片 ra)xΠa,ooⅡbgo uv∈E v∈V Occupancy ratio: PrX~uT[Xv=0G] T PIX~uT [Xo=1] Oi d R= bo ao0R图 +a01 i=1 +a11 a Mobius transformation
Ti v vi T 2-spin system: Occupancy ratio: Tree Recurrence i R T = b0 b1 Y d i=1 a00Ri Ti + a01 a10Ri Ti + a11 R T , PrX⇠µT [Xv = 0 | ] PrX⇠µT [Xv = 1 | ] A = a00 a01 a10 a11 b = b0 b1 µT () / Y uv2E a(u),(v) Y v2V b(v) a Möbius transformation