The question asks whether the Langevin or Glauber-type dynamics for the orthogonally-invariant Sherrington-Kirkpatrick spin glass model, where the interaction matrix is drawn from an orthogonally invariant ensemble rather than a Wigner matrix, exhibit a cutoff phenomenon as the system size grows large. More precisely, one wants to know whether the total variation distance between the law of the dynamics at time t and its stationary measure drops sharply from near 1 to near 0 in a window that is small relative to the mixing time, and to identify both the mixing time and the window size in terms of the spectral data of the interaction matrix and the temperature parameter. The replica-symmetric free energy established in the companion paper gives a precise thermodynamic picture, but the dynamical counterpart, especially the question of abrupt versus gradual mixing, remains entirely open.
This problem is hard for several layered reasons. First, the cutoff phenomenon for spin systems is well understood only in a handful of mean-field models, such as the Curie-Weiss Ising model and the symmetric Potts model, where explicit symmetry makes coupling or spectral arguments tractable. The orthogonally-invariant setting breaks the coordinate permutation symmetry that drives most known cutoff proofs, replacing it with a much richer symmetry group that does not straightforwardly reduce the state space. Second, near the critical temperature the replica-symmetric solution develops singularities, suggesting that the mixing time itself may undergo a phase transition, and standard log-Sobolev or Poincare inequality methods become ineffective precisely in this regime. Third, the dynamical mean-field limit, while yielding a limiting integro-differential equation, does not by itself resolve the pre-limiting cutoff question, which requires uniform control over large but finite N.
Resolving this problem would have broad consequences. A proof of cutoff with an explicit window would provide the first rigorous dynamical counterpart to the replica-symmetric free energy calculation, bridging equilibrium and non-equilibrium behavior in a genuinely new class of disordered systems. It would also clarify whether the spectral distribution of the interaction matrix, rather than just its largest eigenvalue, controls the mixing time, which has implications for algorithm design in high-dimensional inference and sampling. Furthermore, the techniques developed would likely transfer to other orthogonally-invariant models arising in random matrix theory and Bayesian statistics.
This problem sits at the intersection of two of the source papers. The dynamical mean-field limit and replica-symmetric free energy paper provides the equilibrium and limiting-dynamics framework within which the question is naturally posed, supplying candidate mixing-time scales derived from the free energy landscape. The cutoff analysis for the mean-field ferromagnetic Potts model paper provides the methodological template: it shows how to establish cutoff for mean-field spin systems by combining a careful analysis of the magnetization chain near its stationary distribution with sharp concentration estimates. Adapting this program to the orthogonally-invariant SK setting, where the order parameter is a full overlap matrix rather than a scalar magnetization, constitutes the core technical challenge of the open problem.