Glauber dynamics

Remarks

The number $p_v(\sigma)$ is the conditional probability under $\pi$ that the spin at $v$ is $+1$ given the other spins: the two candidates have weights $e^{\pm\beta(JS_v + h)}$. The chain satisfies detailed balance with respect to $\pi$, so $\pi$ is stationary, and since it is irreducible and aperiodic it converges to $\pi$. Example: take a single edge $\set{a, b}$ with $h = 0$, and start from $\sigma = (+1, +1)$. The chain picks $a$ with probability $1/2$ and flips it with probability $1 - p_a = 1/(1 + e^{2\beta J})$. So the move to $(-1, +1)$ has probability $\tfrac12 (1 + e^{2\beta J})^{-1}$, and the reverse move has probability $\tfrac12 (1 + e^{-2\beta J})^{-1}$. Detailed balance holds because $\pi(+,+)/\pi(-,+) = e^{2\beta J}$, which is the reverse move probability divided by the forward one. Non-example: the chain that picks a uniform vertex and always flips its spin fails clause 3. Its stationary law is uniform on $\Omega$, which differs from $\pi$ when $\beta J \ne 0$ and $E$ is non-empty. The same scheme works for any Gibbs measure on a product space; the continuous-time version updates each vertex at the times of an independent rate-one Poisson clock. Glauber's 1963 paper treated the one-dimensional Ising chain with single spin-flip rates of this kind; the Metropolis rule is another common choice with the same stationary law.