  We study the hyperbolic $\Phi^{k+1}_2$-model on the plane. By establishing
coming down from infinity for the associated stochastic nonlinear heat equation
(SNLH) on the plane, we first construct a $\Phi^{k+1}_2$-measure on the plane
as a limit of the $\Phi^{k+1}_2$-measures on large tori. We then study the
canonical stochastic quantization of the $\Phi^{k+1}_2$-measure on the plane
thus constructed, namely, we study the defocusing stochastic damped nonlinear
wave equation forced by an additive space-time white noise (= the hyperbolic
$\Phi^{k+1}_2$-model) on the plane. In particular, by taking a limit of the
invariant Gibbs dynamics on large tori constructed by the first two authors
with Gubinelli and Koch (2021), we construct invariant Gibbs dynamics for the
hyperbolic $\Phi^{k+1}_2$-model on the plane. Our main strategy is to develop
further the ideas from a recent work on the hyperbolic $\Phi^3_3$-model on the
three-dimensional torus by the first two authors and Okamoto (2021), and to
study convergence of the so-called enhanced Gibbs measures, for which coming
down from infinity for the associated SNLH with positive regularity plays a
crucial role. By combining wave and heat analysis together with ideas from
optimal transport theory, we then conclude global well-posedness of the
hyperbolic $\Phi^{k+1}_2$-model on the plane and invariance of the associated
Gibbs measure. As a byproduct of our argument, we also obtain invariance of the
limiting $\Phi^{k+1}_2$-measure on the plane under the dynamics of the
parabolic $\Phi^{k+1}_2$-model.
