No-overlap-gap in the Sherrington–Kirkpatrick model

Determine whether the asymptotic overlap distribution in the Sherrington–Kirkpatrick Ising spin glass with zero external field—i.e., the mixed p-spin model with c_p ≠ 0 only for p = 2 and Hamiltonian H(σ) = N^{-1/2} ∑_{i<j} G_{ij} σ_i σ_j for σ ∈ {+1,−1}^N with i.i.d. standard Gaussian couplings—has no overlap gap; equivalently, prove that the Parisi minimizer γ_* is strictly increasing on [0,1], implying a continuous overlap distribution without gaps.

Background

The paper analyzes algorithms that construct near-ground states in mixed p-spin models and shows that they achieve the ground-state energy if and only if the Parisi optimizer γ_* is strictly increasing, which corresponds to a no-overlap-gap structure. This connects algorithmic performance to the geometry of the Gibbs measure via the replica-symmetry breaking framework.

For the Sherrington–Kirkpatrick model (the p=2 Ising spin glass with zero field), it is widely believed that the overlap distribution exhibits no overlap gap, but a proof is lacking. Establishing this would not only resolve a central structural question in spin glass theory but would also, via the paper’s algorithmic results, imply that polynomial-time algorithms can reach the optimal ground-state energy in this model.