  We perform a comparative Monte Carlo study of the easy-plane deconfined
critical point (DCP) action and its short-range counterpart to reveal close
similarities between the two models for intermediate and strong coupling
regimes. For weak coupling, the structure of the phase diagram depends on the
interaction range: while the short-range model features a tricritical point and
a continuous U(1)xU(1) transition,the long-range DCP action is characterized by
the runaway renormalization flow of coupling into a first (I) order phase
transition. We develop a "numerical flowgram" method for high precision studies
of the runaway effect, weakly I-order transitions, and polycritical points. We
prove that the easy-plane DCP action is the field theory of a weakly I-order
phase transition between the valence bond solid and the easy-plane
antiferromagnet (or superfluid, in particle language) for any value of the weak
coupling strength. Our analysis also solves the long standing problem of what
is the ultimate fate of the runaway flow to strong coupling in the theory of
scalar electrodynamics in three dimensions with U(1)xU(1) symmetry of quartic
interactions
