  We solve the Random Euclidean Matching problem with exponent 2 for the
Gaussian distribution defined on the plane. Previous works by Ledoux and
Talagrand determined the leading behavior of the average cost up to a
multiplicative constant. We explicitly determine the constant, showing that the
average cost is proportional to (log N)^2, where N is the number of points. Our
approach relies on a geometric decomposition allowing an explicit computation
of the constant. Our results illustrate the potential for exact solutions of
random matching problems for many distributions defined on unbounded domains on
the plane.
