= Hard-core model
{title2=$Z_G(\lambda)$}
The hard-core model on a finite <graph> assigns probability to <independent sets> according to an activity $\lambda>0$:
$$
\Pr(I)=\frac{\lambda^{|I|}}{Z_G(\lambda)},\qquad Z_G(\lambda)=\sum_{J\text{ independent}}\lambda^{|J|}.
$$
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.
Back to article page
