  A model of random plane partitions which describes five-dimensional
$\mathcal{N}=1$ supersymmetric SU(N) Yang-Mills is studied. We compute the wave
functions of fermions in this statistical model and investigate their
thermodynamic limits or the semi-classical behaviors. These become of the WKB
type at the thermodynamic limit. When the fermions are located at the main
diagonal of the plane partition, their semi-classical wave functions are
obtained in a universal form. We further show that by taking the
four-dimensional limit the semi-classical wave functions turn to live on the
Seiberg-Witten curve and that the classical action becomes precisely the
integral of the Seiberg-Witten differential. When the fermions are located away
from the main diagonal, the semi-classical wave functions depend on another
continuous parameter. It is argued that they are related with the wave
functions at the main diagonal by the renormalization group flow of the
underlying gauge theory.
