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A variance inequality for spin-flip systems is obtained using comparativelyweaker knowledge of relaxation to equilibrium based on coupling estimates forsingle site disturbances. We obtain variance inequalities interpolating betweenthe Poincar\'e inequality and the uniform variance inequality, and a generalweak Poincar\'e inequality. For monotone dynamics the variance inequality canbe obtained from decay of the autocorrelation of the spin at the origin, i.e.,from that decay we conclude decay for general functions. This method is thenapplied to the low temperature Ising model, where the time-decay of theautocorrelation of the origin is extended to arbitrary quasi-local functions.

A Variance Inequality for Glauber Dynamics applicable to High and Low Temperature Regimes