  We study an extension of the Seiberg-Witten theory of $5d$ $\mathcal{N}=1$
supersymmetric Yang-Mills on $\mathbb{R}^4 \times S^1$. We investigate
correlation functions among loop operators. These are the operators analogous
to the Wilson loops encircling the fifth-dimensional circle and give rise to
physical observables of topological-twisted $5d$ $\mathcal{N}=1$ supersymmetric
Yang-Mills in the $\Omega$ background. The correlation functions are computed
by using the localization technique. Generating function of the correlation
functions of U(1) theory is expressed as a statistical sum over partitions and
reproduces the partition function of the melting crystal model with external
potentials. The generating function becomes a $\tau$ function of 1-Toda
hierarchy, where the coupling constants of the loop operators are interpreted
as time variables of 1-Toda hierarchy. The thermodynamic limit of the partition
function of this model is studied. We solve a Riemann-Hilbert problem that
determines the limit shape of the main diagonal slice of random plane
partitions in the presence of external potentials, and identify a relevant
complex curve and the associated Seiberg-Witten differential.
