  We introduce and study a one parameter deformation of the polynuclear growth
(PNG) in $(1+1)$-dimensions, which we call the $t$-PNG model. It is defined by
requiring that, when two expanding islands merge, with probability $t$ they
sprout another island on top of the merging location. At $t=0$, this becomes
the standard (non-deformed) PNG model that, in the droplet geometry, can be
reformulated through longest increasing subsequences of uniformly random
permutations or through an algorithm known as patience sorting. In terms of the
latter, the $t$-PNG model allows errors to occur in the sorting algorithm with
probability $t$.
  We prove that the $t$-PNG model exhibits one-point Tracy-Widom GUE
asymptotics at large times for any fixed $t\in [0,1)$, and one-point
convergence to the narrow wedge solution of the Kardar-Parisi-Zhang (KPZ)
equation as $t$ tends to $1$. We further construct distributions for an
external source that are likely to induce Baik-Ben Arous-Peche type phase
transitions. The proofs are based on solvable stochastic vertex models and
their connection to the determinantal point processes arising from Schur
measures on partitions.
