Here exponentially localized can be thought in a non-rigorous manner as a measure that is mostly supported on a sparse number of nodes.

Some intuition can gained by thinking about a diffusion process, e.g., we know that in presence of a "confining potential" e.g., $V(x)\approx x^2/2$, the Fokker-Planck equation in 1D will have an exponentially localized stationary distribution (Boltzmann distribution).

$\dfrac{\partial P}{\partial t}-\partial_x(P\partial_x V)=\sigma\partial_{xx}P$

Is there corresponding Markov Chain literature that also studies such localization properties, e.g. providing conditions on the Markov chain transition matrix that lead to such localization ?