# Random walk through Ising model

Ising model consists of up spin and down spin or empty/filled space. Can we model random walk for different densities of packing through the Ising model?

• What do you mean by "Can we model random walk for different densities of packing through ising model"? Mar 9 '17 at 16:17
• Hello Yvan , i am working on tipping points (phase transition) in ecological systems such as lake system using ising model. Here i am focusing on flow dynamics through vegetation .I want to simulate particle velocity using ising model for different packing of spins (vegetation ) .Hope , i made it clear Mar 9 '17 at 23:14

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 $$w\times \infty$$ 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 $$M\psi = \lambda \psi$$, 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: 