We can derive the correlation time by observing that our Metropolis algorithm is based on a random walk in the space of all possible spin configurations. Our probability distribution function \( \mathbf{\hat{w}}(t) \) after a given number of time steps \( t \) could be written as

$$ \mathbf{\hat{w}}(t) = \mathbf{\hat{W}^t\hat{w}}(0), $$

with \( \mathbf{\hat{w}}(0) \) the distribution at \( t=0 \) and \( \mathbf{\hat{W}} \) representing the transition probability matrix. We can always expand \( \mathbf{\hat{w}}(0) \) in terms of the right eigenvectors of \( \mathbf{\hat{v}} \) of \( \mathbf{\hat{W}} \) as

$$ \mathbf{\hat{w}}(0) = \sum_i\alpha_i\mathbf{\hat{v}}_i, $$

resulting in

$$ \mathbf{\hat{w}}(t) = \mathbf{\hat{W}}^t\mathbf{\hat{w}}(0)=\mathbf{\hat{W}}^t\sum_i\alpha_i\mathbf{\hat{v}}_i= \sum_i\lambda_i^t\alpha_i\mathbf{\hat{v}}_i, $$

with \( \lambda_i \) the \( i^{\mathrm{th}} \) eigenvalue corresponding to the eigenvector \( \mathbf{\hat{v}}_i \).