Transfer matrix for a classical spin chain ($Z_N=\mathrm{tr}{W}^N$)

For a nearest-neighbour classical chain with finitely many spin states and dimensionless bond energy $E(s,t)$, its transfer matrix is $W_{st}=e^{-E(s,t)}$. On-site energies are divided between adjacent bonds. Periodic boundary conditions give partition function $Z_N=\mathrm{tr}{W}^N$ by direct multiplication and summation over spin labels. For a strictly positive symmetric matrix, the Perron–Frobenius theorem gives ${\lim }_{N\to \infty }{N}^{-1}\log Z_N=\log {\lambda }_{\max }$. A diagonal local observable $O$ has expectation $\mathrm{tr}(O{W}^N)/Z_N$, tending to $v_{\max }^{T}O{v}_{\max }$ for a normalized dominant eigenvector.