Sample the target measure and measure the correlations

At a finite lattice regulator with nonnegative target weight, normalize π(U) proportional to exp(−Sg) times the included fermion determinant. Integrals over configurations use the specified product Haar measure. A transition kernel preserves π if integrating over the starting configuration produces π again. Detailed balance equates probability flow in each direction and is a sufficient construction for stationarity, although globally balanced algorithms need not satisfy it pair by pair.

The familiar Metropolis rule min[1,exp(−ΔS)] assumes a proposal symmetric under reversal with respect to the same measure. Asymmetric proposals require the appropriate proposal-density ratio. Rejected moves count as repeated configurations and must remain in the chain. Keeping accepted proposals only generally changes the sampled distribution, because states with different rejection rates would then be weighted incorrectly. This bookkeeping issue is independent of whether the action evaluation itself is accurate.

Convergence from an arbitrary start requires suitable irreducibility and aperiodicity, or the corresponding ergodic conditions in the continuous state space. A deterministic alternating chain can have a stationary distribution but fail to approach it from one starting state. Near a continuum limit, rare transitions between topological sectors may create very long relaxation times. A short apparently stable record of common observables does not establish that every slow mode has equilibrated.

Discarding a chosen warmup period addresses initialization only if that period is long compared with relevant relaxation scales. Independent starts, sector histories, and observable-specific autocorrelations provide complementary checks. Thinning can reduce stored correlations but discards information and does not magically create more independent data than were generated. The Monte Carlo update index is algorithmic time; it is not Euclidean propagation time used to extract a hadron energy.

For a sufficiently mixing chain with summable correlations, the sample-mean variance is approximately the single-configuration variance divided by the effective count. Estimating the correlation sum itself requires a window choice and enough data to see its tail. Blocking or correlated resampling should respect those scales. A reported standard error with an unobserved slow tail can be optimistic even when its numerical computation is exact.

The Markov lab makes finite convergence visible. Reset and set p=0.2 and q=0.4, then Step twice; predict state-1 probability 0.28 and stationary value 1/3. Reset and set p=0.4 and q=0.2, then Step twice; predict 0.56 and stationary value 2/3. The lab evolves exact probability distributions from state 0 rather than generating a random chain. It has no gauge configurations, measured autocorrelation error bars, or continuum extrapolation.