Hard-core model ($Z_G(\lambda )$)

The hard-core model on a finite graph assigns probability to independent sets according to an activity $\lambda >0$:An occupied vertex excludes occupation of every neighbor, which is the hard constraint. At $\lambda =1$, all independent sets are equally likely, not all vertex subsets.

For one edge with vertices $u,v$, the allowed sets are $\varnothing ,\{u\},\{v\}$, giving $Z=1+2\lambda$ and occupation probability $\lambda /(1+2\lambda )$ for each vertex. The set $\{u,v\}$ has probability zero. Increasing activity favors larger allowed sets while retaining the exclusion rule.

The finite partition function satisfies $Z_G=Z_{G-v}+\lambda Z_{G-N[v]}$: split independent sets according to whether they contain $v$, deleting its closed neighborhood in the occupied case. This recurrence connects the statistical-mechanical model to counting independent sets.