att ices ▁What ▁follows ▁is ▁a ▁formal ▁definition ▁for ▁the ▁special ▁case ▁of ▁a ▁random ▁field ▁on ▁a ▁lattice . ▁The ▁idea ▁of ▁a ▁Gib bs ▁measure ▁is , ▁however , ▁much ▁more ▁general ▁than ▁this . ▁ ▁The ▁definition ▁of ▁a ▁Gib bs ▁random ▁field ▁on ▁a ▁lattice ▁requires ▁some ▁termin ology : ▁▁ ▁The ▁lattice : ▁A ▁count able ▁set ▁. ▁ ▁The ▁single - sp in ▁space : ▁A ▁probability ▁space ▁. ▁ ▁The ▁configuration ▁space : ▁, ▁where ▁ ▁and ▁. ▁ ▁Given ▁a ▁configuration ▁ ▁and ▁a ▁subset ▁, ▁the ▁restriction ▁of ▁ ▁to ▁ ▁is ▁. ▁If ▁ ▁and ▁, ▁then ▁the ▁configuration ▁ ▁is ▁the ▁configuration ▁whose ▁restrictions ▁to ▁ ▁and ▁ ▁are ▁ ▁and ▁, ▁respectively . ▁ ▁The ▁set ▁ ▁of ▁all ▁finite ▁subsets ▁of ▁. ▁ ▁For ▁each ▁subset ▁, ▁ ▁is ▁the ▁ ▁- algebra ▁generated ▁by ▁the ▁family ▁of ▁functions ▁, ▁where ▁. ▁The ▁union ▁of ▁these ▁- al gebras ▁as ▁ ▁var ies ▁over ▁ ▁is ▁the ▁algebra ▁of ▁cyl inder ▁sets ▁on ▁the ▁lattice . ▁ ▁The ▁potential : ▁A ▁family ▁ ▁of ▁functions ▁ ▁such ▁that ▁ ▁For ▁each ▁ ▁is ▁- me as urable , ▁meaning ▁it ▁depends ▁only ▁on ▁the ▁restriction ▁ ▁( and ▁does ▁so ▁meas ur ably ). ▁ ▁For ▁all ▁ ▁and ▁, ▁the ▁following ▁series ▁exists : ▁ ▁We ▁interpret ▁▁ ▁as ▁the ▁contribution ▁to ▁the ▁total ▁energy ▁( the ▁Hamiltonian ) ▁associated ▁to ▁the ▁interaction ▁among ▁all ▁the ▁points ▁of ▁finite