  We obtain scaling and local limit results for large random Young tableaux of
fixed shape $\lambda^0$ via the asymptotic analysis of a determinantal point
process due to Gorin and Rahman (2019). More precisely, we prove:
  (1) an explicit description of the limiting surface of a uniform random Young
tableau of shape $\lambda^0$, based on solving a complex-valued polynomial
equation;
  (2) a simple criteria to determine if the limiting surface is continuous in
the whole domain;
  (3) and a local limit result in the bulk of a random Poissonized Young
tableau of shape $\lambda^0$.
  Our results have several consequences, for instance: they lead to explicit
formulas for the limiting surface of $L$-shaped tableaux, generalizing the
results of Pittel and Romik (2007) for rectangular shapes; they imply that the
limiting surface for $L$-shaped tableaux is discontinuous for almost-every
$L$-shape; and they give a new one-parameter family of infinite random Young
tableaux, constructed from the so-called random infinite bead process.
