= Transfer matrix for a classical spin chain
{title2=$Z_N=\operatorname{tr}W^N$}
= Spin-chain transfer matrix
{synonym}
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> $Z_N=\operatorname{tr}W^N$ 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> $\operatorname{tr}(OW^N)/Z_N$, tending to $v_{\max}^TOv_{\max}$ for a normalized dominant <eigenvector>.
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