undirected graph G=V,E) monomer-dimer model:set of all matchings M(G) partition function Z,(G)=∑XM M∈M(G) λXMI Gibbs distribution (M)= Zλ(G) hardcore model: set of all independent sets Z(G) partition function Z(G)=>X四 I∈I(G) λI Gibbs distribution ()= Z,(G)Z￾(G) = X I2I(G) ￾|I| Z￾(G) = X M2M(G) ￾|M| M(G) I(G) monomer-dimer model: hardcore model: undirected graph G = (V,E) set of all matchings partition function partition function set of all independent sets µ(I) = ￾|I| Z￾(G) µ(M) = ￾|M| Z￾(G) Gibbs distribution Gibbs distribution