Balanced multi-species spin glasses

We identify a special class of multi-species spin glass models: ones in which the species proportions serve to ''balance'' out the interaction strengths. For this class, we prove a free energy lower bound that does not require any convexity assumption, and applies to both Ising and spherical models. The lower bound is the free energy of a single-species model whose $p$-spin inverse-temperature parameter is exactly the variance of the $p$-spin component of the multi-species Hamiltonian. For the Ising case, this generalizes an inequality recently found by Issa in the context of vector spin models. We further demonstrate that this lower bound is actually an equality in many cases, including: at high temperatures for all models, at all temperatures for convex models, and at zero temperature for pure bipartite spherical models. When translated to a statement about the injective norm of a nonsymmetric Gaussian tensor, our lower bound matches an upper bound recently established by Dartois and McKenna.