Quiver asymptotics and amoeba: Instantons on toric Calabi-Yau divisors

The BPS bound states of D4-D2-D0 branes on the noncompact divisors of Calabi-Yau threefolds and the instantons in the dual quiver gauge theories are previously studied using two-dimensional crystal melting model and dimer model, [T. Nishinaka et al., J. High Energy Phys. 5 (2014) 139]. Using the tropical geometry associated with the toric quiver, we study the asymptotic of the quiver gauge theory to compute some of their thermodynamic observables and extract the phase structure. We obtain that the thermodynamic observables such as free energy, entropy, and growth rate are explicitly derived from the limit shape of the crystal model, the boundary of the amoeba and its Harnack curve characterization. Furthermore, we observe that there is a Hagedorn phase transition in the instanton sector inferred from the Gumbel distribution of the fluctuations in the crystal model. We present explicit computations of the results in concrete examples of C-3, conifold, local P-1 x P-1 and local P-2 quivers.