  The theory of Markov processes on the square lattice has been given recently
by the second author a new algebraic description in terms of operads. In
particular, this new approach allows for a nice description of invariant
boundary conditions and infinite-volume Gibbs measures. This theory comes with
new algebraic objects which have not been constructed on any non trivial model
yet. In this article, the main objective is to exhibit and understand these
structures in the particular case of Gaussian Markov fields on the
two-dimensional square lattice. This article, in the Gaussian framework, is the
first time where all the operadic constructions -- products and eigen-elements
up to morphisms -- introduced by the second author are defined rigorously. We
also relate these constructions to more classical approach such as the transfer
matrix of statistical mechanics and the Fourier transform. The description of
half-strips and corners is new and requires the introduction of new operations
such as folding. From the probabilistic point of view, we also show that the
operadic products on the boundaries are not easily defined and most operations
are lifted to the level of parameter spaces, here quadratic forms through Schur
complements.
