Correlation function for the Grid-Poisson Euclidean matching on a line and on a circle

We compute the two-point correlation function for spin configurations which are obtained by solving the Euclidean matching problem, for one family of points on a grid and the second family chosen uniformly at random, when the cost depends on a power p of the Euclidean distance. We provide the analytic solution in the thermodynamic limit, in a number of cases (p > 1 open b.c. and p = 2 periodic b.c., both at criticality) and analyse numerically other parts of the phase diagram.