Random walk through Ising model

There are ways to map the magnetization distribution problem of the Ising model to persistent random walks. The relevant references are:

R. Garcia-Pelayo, "Distribution of magnetization in the finite Ising chain," Journal of Mathematical Physics 50, 013301 (2009)

and

T. Antal, M. Droz, and Z. Racz, "Probability distribution of magnetization in the one-dimensional Ising model: effects of boundary conditions," Journal of Physics A: Mathematics and General 37, 1465 (2004)

Yes, we can imagine going through 1D Ising model as random walk, analogous to Maximal Entropy Random Walk, e.g. to model 2D Ising $wtimes infty$ as 1D for width $w$ slices ( https://arxiv.org/pdf/1912.13300 ).

For $M_{ij}=exp(-beta E_{ij})$ transfer matrix, where $E_{ij}$ is energy of $i-j$ edge (e.g. negative for the same spins in ferromagnet), you need to find dominant eigenvector $Mpsi = lambda psi$, then

stationary probability distribution is $Pr(i)propto (psi_i)^2$,

Stochastic matrix: $S_{ij}=Pr(x_t=j|x_{t-1}=i)= frac{M_{ij}}{lambda} frac{psi_j}{psi_i}$.

Sketch of derivation: