  The large N generating functions for the counting of chiral operators in
$\mathcal{N}=1$, four-dimensional quiver gauge theories have previously been
obtained in terms of the weighted adjacency matrix of the quiver diagram. We
introduce the methods of multi-variate asymptotic analysis to study this
counting in the limit of large charges. We describe a Hagedorn phase transition
associated with this asymptotics, which refines and generalizes known results
on the 2-matrix harmonic oscillator. Explicit results are obtained for two
infinite classes of quiver theories, namely the generalized clover quivers and
affine $\mathbb{C}^3/\hat{A}_n$ orbifold quivers.
