LAUSR

Abstract The BPS sector in $$ \mathcal{N}=2 $$ N = 2 , four-dimensional toric quiver gauge theories has previously been studied using the crystal melting model and dimer model. We introduce the Mahler measure associated with the statistical dimer model to study the large N limit of these quiver gauge theories. In this limit, the 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 the profile function, entropy density, BPS growth rate and phase structure of quivers are presented in concrete examples of the clover ℂ3, and resolved conifold $$ \mathcal{C} $$ C quivers. Finally, BPS growth rates of Hirzebruch $$ \mathbb{F} $$ F 0, and ℂ3 /ℤ2 × ℤ2 orbifold quivers are obtained and a possible interpretation of the results for certain BPS black holes is discussed.