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

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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.