Quiver superconformal index and giant gravitons: asymptotics and expansions

We study asymptotics of the $d=4$, N=1 superconformal index for toric quiver gauge theories. Using graph-theoretic and algebraic factorization techniques, we obtain a cycle expansion for the large-N index in terms of the R-charge-weighted adjacency matrix. Applying saddle-point techniques at the on-shell R-charges, we determine the asymptotic degeneracy in the univariate specialization for Âₘ, and along the main diagonal for the bivariate index for N=4 and Â₃. In these cases we find ln |cₙ| ~ γn1/2+ βln n + α (Hardy-Ramanujan type). We also identify polynomial growth for $dP3$, Y3,3 and Yp,0, and give numerical evidence for $γ$ in further Yp,p examples. Finally, we generalize Murthy's giant graviton expansion via the Hubbard-Stratonovich transformation and Borodin-Okounkov formula to multi-matrix models relevant for quivers.