Consider an instance of the Ising model, with $N$ number of spins on a 2D square lattice (or any other 2D structure) wrapped into a torus to avoid boundary conditions (in other words, periodic boundary conditions) with nearest neighbor coupling. I am sampling from the Boltzmann probability distribution: $$\mu(s)=\frac{e^{-E(s)/T}}{Z}$$ using Monte Carlo Markov chains (MCMC). $Z$ is the partition function, $s$ is a spin configuration ($s\in S$ with $S$ state space containing all the possible spin configurations for the $N$ spins system), and $E(s)$ is the energy whose definition depends on the specific instance of the model.

Consider the following transition matrix P representing the MC process: $$P(s'|s)=A(s'|s)C(s'|s)$$ where $C$ is the proposal strategy and $A$ is the acceptance probability (Metropolis-Hastings):$$A(s'|s)=min(1, \frac{\mu(s')C(s|s')}{\mu(s)C(s'|s)})$$ $C$ can be any (good) proposal distribution such that $P$ satisfies the detailed balance condition: $$P(s'|s)\mu(s)=P(s|s')\mu(s')$$

The MCMC transition matrix $P$ defines the spectral gap:


where $\lambda$ is the second largest eigenvalue of $P$. The spectral gap is an important measure of how fast the Markov chain converges to the target probability distribution (Boltzmann probability in this case). It is also directly related to other important quantities as the relaxation time and the exponential autocorrelation time.

Given a certain $C$, I am trying to understand the relationship between the system dimension, the number of spins $N$, and the spectral gap $\delta$. How does $\delta$ scale with respect to the number of spins $N$? Could someone please suggest relevant literature regarding this relationship?

Any information about the relationship between the relaxation time or the autocorrelation time and the number of spins $N$ would be equally helpful.

Information about the scaling of the mixing time with respect to the number of spins $N$ could help as well. The mixing time is not directly related to the spectral gap but it can be used to calculate upper and lower bounds.



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