Finite-size critical-exponent fitting

For the Ising source H appearing in the dimensionless action, differentiating the mean magnetization density gives the susceptibility N times the variance of m. If the source instead is the energy-valued physical field h, an extra inverse-temperature factor enters. At exact zero field in a finite symmetric ensemble, the signed mean magnetization vanishes, so absolute magnetization or its root-mean-square can serve as a nonzero finite-size amplitude. A susceptibility with absolute magnetization subtracted is another observable and must be labeled rather than silently substituted.

At a continuous transition in the ordinary scaling regime, susceptibility grows with exponent gamma over nu, while absolute magnetization shrinks with the magnetization exponent divided by nu. A dimensionless Binder slope grows with inverse nu. We write the magnetization exponent as beta mag to avoid confusing it with inverse temperature. These laws assume the correct critical coupling, fixed shape and boundaries, and appropriate control of irrelevant fields. First-order coexistence, upper-critical-dimensional logarithms, or dangerous irrelevant variables can require modified forms.

A ratio between sizes L and sL cancels a leading amplitude. Taking its logarithm and dividing by logarithm of s gives an effective exponent. For an exact power law this equals the exponent at every size; with a correction proportional to L raised to minus omega, the quotient drifts. A regular additive background can produce a different drift, especially when the singular contribution is not yet dominant. A visually stable pair of adjacent ratios can result from cancellation between corrections.

Evaluate several sizes and vary the minimum size retained in the fit. Fit corrections only when the data can constrain their amplitudes and powers; otherwise an apparently flexible model can conceal the absence of an asymptotic regime. Reweighting to a pseudocritical crossing can reduce some tuning uncertainty but introduces a size-dependent coupling trajectory. Its scaling interpretation must match the chosen estimator. A slope exponent is sensitive to how the reduced coupling is defined, although a constant coupling rescaling changes its amplitude rather than its asymptotic power.

Measurements at multiple couplings derived from the same trajectory are correlated. Bootstrap or jackknife resamples should repeat the full moment construction, reweighting, coupling selection, and exponent fit, using blocks that resolve the slowest relevant measurement mode. A poorly estimated covariance matrix can make a correlated fit unstable; blindly inverting it does not make the analysis more reliable. Independent replicas and an explicit covariance-treatment choice help separate statistical limitations from an inadequate scaling model.

Scaling relations provide consistency checks only under their hypotheses. For short-range systems below the upper critical dimension with ordinary hyperscaling, twice beta mag plus gamma equals d times nu. Boundary conditions, anisotropy, long-range forces, and finite imaginary-time extent in quantum models require further analysis. Agreement of one fitted ratio with a famous number cannot establish a universality class. Compare several dimensionless quantities and exponents, while preserving a record of discarded sizes and fit choices. The quadratic instrument tests one supplied power and performs no exponent inference.