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 $\mathbf{\hat{v}}_i$.