  We study probabilistic and combinatorial aspects of natural volume-and-trace
weighted plane partitions and their continuous analogues. We prove asymptotic
limit laws for the largest parts of these ensembles in terms of new and known
hard- and soft-edge distributions of random matrix theory. As a corollary we
obtain an asymptotic transition between Gumbel and Tracy--Widom GUE
fluctuations for the largest part of such plane partitions, with the continuous
Bessel kernel providing the interpolation. We interpret our results in terms of
two natural models of directed last passage percolation (LPP): a discrete
$(\max, +)$ infinite-geometry model with rapidly decaying geometric weights,
and a continuous $(\min, \cdot)$ model with power weights.
