  Large surveys of star-forming regions have unveiled power-law correlations
between the stellar mass and the disc parameters, such as the disc mass
$M_{\mathrm{d}} \propto {M_{\star}}^{\lambda_{\mathrm{m}}}$ and the accretion
rate $\dot M \propto {M_{\star}}^{\lambda_{\mathrm{acc}}}$. The observed slopes
appear to be increasing with time, but the reason behind the establishment of
these correlations and their subsequent evolution is still uncertain. We
conduct a theoretical analysis of the impact of viscous evolution on power-law
initial conditions for a population of protoplanetary discs. We find that, for
evolved populations, viscous evolution enforces the two correlations to have
the same slope, $\lambda_{\mathrm{m}}$ = $\lambda_{\mathrm{acc}}$, and that
this limit is uniquely determined by the initial slopes $\lambda_{\mathrm{m},
0}$ and $\lambda_{\mathrm{acc}, 0}$. We recover the increasing trend claimed
from the observations when the difference in the initial values, $\delta_0 =
\lambda_{\mathrm{m}, 0} - \lambda_{\mathrm{acc}, 0}$, is larger than $1/2$;
moreover, we find that this increasing trend is a consequence of a positive
correlation between the viscous timescale and the stellar mass. We also present
the results of disc population synthesis numerical simulations, that allow us
to introduce a spread and analyse the effect of sampling, which show a good
agreement with our analytical predictions. Finally, we perform a preliminary
comparison of our numerical results with observational data, which allows us to
constrain the parameter space of the initial conditions to
$\lambda_{\mathrm{m}, 0} \in [1.2, 2.1]$, $\lambda_{\mathrm{acc}, 0} \in [0.7,
1.5]$.
