Self-overlap correction simplifies the Parisi formula for vector spins

Abstract: We propose a simpler approach to identifying the limit of free energy in a vector spin glass model by adding a self-overlap correction to the Hamiltonian. This avoids constraining the self-overlap and allows us to identify the limit with the classical Parisi formula, similar to the proof for scalar models with Ising spins. For the upper bound, the correction cancels self-overlap terms in Guerra's interpolation. For the lower bound, we add an extra perturbation term to make the self-overlap concentrate, a technique already used in [Probab. Math. Phys., 2(2):281-339, Ann. Inst. Henri Poincar\'e Probab. Stat., 59(3):1143-1182] to ensure the Ghirlanda-Guerra identities. We then remove the correction using a Hamilton-Jacobi equation technique, which yields a formula similar to that in [Ann. Probab., 46(2):865-896]. Additionally, we sketch a direct proof of the main result in [Electron. J. Probab. 25(23):1-17].