ρDistance and Distortion-Rate Function of Random Fields

We introduce the notation of ρ-distance of two random fields, and show the validness of the definition. Some lower and upper bounds are derived and applied to evaluate some examples. In particular, we show that the ρ-distance of the two extreme random fields μ₊,μ_- of 2-D Ising model on Z² is larger than 1/3 in the case of phase transition. We also introduce the notation of distortion-rate function of random fields. We pursue the relation betweenρ-distance and distortion-rate function, and obtain some interesting properties that are useful in the proof of coding theorem.