Histogram reweighting and overlap limits

At zero external field, write B for the sum of neighboring spin products in the once-counted graph convention. The target action is minus K times B. Increasing K therefore favors larger B, so the reweighting factor carries a positive coupling difference in its exponent. Dividing the reweighted observable average by the average weight cancels the unknown ratio of partition functions. An additive action constant also cancels from this normalized estimator if treated consistently in numerator and denominator.

A histogram of B is sufficient for observables that depend only on B; more general observables need their joint measurements or conditional averages in each bin. Reweighting a nonlinear statistic requires reweighting its primary moments and then reconstructing the statistic. For example, averaging a configuration-level Binder-like ratio is not the same operation as forming a ratio of reweighted ensemble moments. The self-normalized finite-sample estimator generally has ratio bias, even though the infinite-ensemble identity is exact.

For positive weights, a commonly quoted effective size compares the square of their sum with the sum of their squares. Equal weights give the number of entries, while a single dominant entry gives approximately one. This measures empirical concentration among observed configurations. It is not the autocorrelation-based effective size for an arbitrary reweighted observable, and multiplying two unrelated effective-size diagnostics does not generally yield a justified error estimate. Joint blocked analysis of numerator and denominator remains necessary.

A large empirical effective size can still be misleading if every recorded configuration misses the region that dominates the new target. No manipulation of existing weights can reveal that absent support. In a well-behaved single energy peak, a rough local warning scale is coupling change times the standard deviation of B becoming appreciable, but this is not a universal sufficiency bound. Inspect overlapping histograms from additional couplings and independent runs. Multi-histogram combinations extend coverage only when their sampled regions connect adequately.

When a formal target weight has a sign or complex phase, sample its absolute magnitude only if that defines a normalizable accessible positive measure. The observable then becomes a ratio containing the fluctuating phase in both numerator and denominator. A small average phase makes the ratio sensitive to cancellation and correlated noise. The positive-weight concentration formula cannot simply be applied to signed weights as if they were independent probabilities. Complex-Langevin or other alternative methods require separate correctness arguments and are not implied by this identity.

Even independent phase measurements can require a large sample to resolve a small denominator. For a sign taking values plus or minus one, its variance is one minus the square of its mean. Keeping a fixed relative uncertainty as that mean shrinks costs roughly its inverse square in independent measurements. In some systems the mean phase also falls exponentially with volume, making the expense severe. This is a statement about a specified representation and regime, not a proof that every physical model has an identical unavoidable numerical cost.