  K-essence is a minimally-coupled scalar field whose Lagrangian density
$\mathcal{L}$ is a function of the field value $\phi$ and the kinetic energy
$X=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi$. In the thawing scenario, the
scalar field is frozen by the large Hubble friction in the early universe, and
therefore initial conditions are specified. We construct thawing k-essence
models by generating Taylor expansion coefficients of $\mathcal{L}(\phi, X)$
from random matrices. From the ensemble of randomly generated thawing k-essence
models, we select dark energy candidates by assuming negative pressure and
non-growth of sub-horizon inhomogeneities. For each candidate model the dark
energy equation of state function is fit to the Chevallier-Polarski-Linder
parameterization $w(a) \approx w_0+w_a(1-a)$, where $a$ is the scale factor.
The thawing k-essence dark models distribute very non-uniformly in the $(w_0,
w_a)$ space. About 90\% models cluster in a narrow band in the proximity of a
slow-roll line $w_a\approx -1.42
\left(\frac{\Omega_m}{0.3}\right)^{0.64}(1+w_0)$, where $\Omega_m$ is the
present matter density fraction. This work is a proof of concept that for a
certain class of models very non-uniform theoretical prior on $(w_0, w_a)$ can
be obtained to improve the statistics of model selection.
