Imaging geometry through dynamics: the observable representation

For many stochastic processes there is an underlying coordinate space, $V$, with the process moving from point to point in $V$ or on variables (such as spin configurations) defined with respect to $V$. There is a matrix of transition probabilities (whether between points in $V$ or between variables defined on $V$) and we focus on its slow'' eigenvectors, those with eigenvalues closest to that of the stationary eigenvector. These eigenvectors are the observables,‘’ and they can be used to recover geometrical features of $V$.