Threshold for sync‑greedy cycles vs fixed points tends to zero with system size

Prove that the finite‑size threshold κ(N) in the sync‑greedy dynamics of the Sherrington–Kirkpatrick (SK) model—defined as the value of the parameter κ for which the probability of converging to fixed points (1‑cycles) equals the probability of converging to 2‑cycles—satisfies lim_{N→∞} κ(N) = 0.

Background

For the sync‑greedy rule σ_GR(x,h;κ)=x·sign(xh+κ), simulations reveal a size‑dependent threshold κ(N) separating regimes dominated by 2‑cycles (small κ) and 1‑cycles (larger κ).

Empirical data up to large N suggest κ(N) decays approximately like 1/log N, indicating a possible vanishing threshold in the thermodynamic limit.