Correlation lengths in the language of computable information

Computable information density (CID), the ratio of the length of a losslessly compressed data file to that of the uncompressed file, is a measure of order and correlation in both equilibrium and nonequilibrium systems. Here we show that correlation lengths can be obtained by decimation, thinning a configuration by sampling data at increasing intervals and recalculating the CID. When the sampling interval is larger than the system's correlation length, the data becomes incompressible. The correlation length and its critical exponents are thus accessible with no a priori knowledge of an order parameter or even the nature of the ordering. The correlation length measured in this way agrees well with that computed from the decay of two-point correlation functions ${g}_{2}(r)$ when they exist. But the CID reveals the correlation length and its scaling even when ${g}_{2}(r)$ has no structure, as we demonstrate by ``cloaking'' the data with a Rudin-Shapiro sequence.