Reciprocal Fields: A Model for Random Fields Pinned to Two Boundaries

We define reciprocal fields as a model for random fields pinned to two boundaries, an example of which is the temperature distribution of a thick pipe with appropriate boundary conditions on the inside and outside wall. Previous studies of models for random fields have been restricted to fields pinned to one boundary, called Markov random fields. Our key contribution is to derive from the reciprocal field (pinned to a two boundary hypersurfaces) a statistically equivalent Markov random field (pinned to a one boundary hypersurface). Going from the reciprocal field to a Markov random field is accomplished by an Origami type folding of the index set of the reciprocal field. This allows the use of known tools for Markov random fields to reciprocal fields. As an example, we derive recursive representations and recursive estimators for reciprocal fields that initiate at the two boundaries of a reciprocal field and telescope to any appropriate hypersurface between the two boundaries.