Toric Quiver Asymptotics and Mahler Measure: $\mathcal{N}=2$ BPS States

BPS sector in $\mathcal{N}=2$, four-dimensional toric quiver gauge theories has previously been studied using crystal melting model and dimer model. We introduce the Mahler measure associated to statistical dimer model to study large $N$ limit of these quiver gauge theories. In this limit, generating function of BPS states in a general toric quiver theory is studied and entropy, growth rate of BPS states and free energy of the quiver are obtained in terms of the Mahler measure. Moreover, a possible third-order phase transition in toric quivers is discussed. Explicit computations of profile function, entropy density, BPS growth rate and phase structure of quivers are presented in concrete examples of clover $\mathbb{C}^3$, and resolved conifold $\mathcal{C}$ quivers. Finally, BPS growth rates of Hirzebruch $\mathbb{F}_0$, and $\mathbb{C}^3/ \mathbb{Z}^2\times \mathbb{Z}^2$ orbifold quivers are obtained and a possible interpretation of the results for certain BPS black holes is discussed.