Dana Randall and I find thresholds in the local–global correlations exhibited by the steady-state distributions of a particle system and the dynamics of the Sherrington–Kirkpatrick spin glass.
For a Markov chain with a finite number of states and a unique stationary distribution $\pi$, the local–global correlation is defined as
$$\rho = \mathrm{Corr}(-\log \pi (X), \log q(X)),$$
where $q(x)$ is the exit rate of state $x$ and $X$ is a uniformly random state.
It is important to identify classes of Markov chains for which $\rho$ is close to $1$, because these Markov chains satisfy an approximate analogue of the Boltzmann distribution. In this analogy, the logarithm of the exit rate plays the role of energy, and the states with the highest values of $\pi$ are simply those with the lowest exit rates. In other words, when $\rho \approx 1$, there is a simple explanation of the order that $\pi$ exhibits.
The challenge is that $\rho$ is generally difficult to calculate, so our recent work resorts to bounding below its expected value under a certain distribution on Markov chains. In a new paper, Dana and I provide the first explicit estimate of $\rho$ in a model system of interest, namely, dynamics of the Sherrington–Kirkpatrick spin glass on the $N$-dimensional hypercube ${-1,1}^N$. We find that, as a parameter $\lambda \in [0,1]$ of the dynamics varies, the correlation $\rho$ jumps from $-1$ to $1$. More precisely, we prove that
$$\rho(\lambda) = \mathrm{sign}(\lambda-\lambda_\ast) \left(1 + \mathrm{error} \right),$$
for $\lambda \neq \lambda_\ast = (1 - 4/N)/(2 - 4/N)$ and where the error is negligible at sufficiently high temperatures, depending on $N$ and $|\lambda-\lambda_\ast|$.
Our hope is that this paper makes the notion of local–global correlation more broadly accessible by presenting a concrete estimate of $\rho$.