If we assume that $\lambda_0$ is the largest eigenvector we see that in the limit $t\rightarrow \infty$, $\mathbf{\hat{w}}(t)$ becomes proportional to the corresponding eigenvector $\mathbf{\hat{v}}_0$. This is our steady state or final distribution.

We can relate this property to an observable like the mean energy. With the probabilty $\mathbf{\hat{w}}(t)$ (which in our case is the squared trial wave function) we can write the expectation values as

$$\langle \mathbf{M}(t) \rangle = \sum_{\mu} \mathbf{\hat{w}}(t)_{\mu}\mathbf{M}_{\mu},$$

or as the scalar of a vector product

$$\langle \mathbf{M}(t) \rangle = \mathbf{\hat{w}}(t)\mathbf{m},$$

with $\mathbf{m}$ being the vector whose elements are the values of $\mathbf{M}_{\mu}$ in its various microstates $\mu$.