Closed-form ensemble-averaged spread complexity for two-level Gaussian random matrices at generic temperature

Derive closed-form analytic expressions, valid for generic inverse temperature β, for the ensemble-averaged spread complexity ⟨K(t, β)⟩ and its long-time average ⟨K̄(β)⟩ in the two-dimensional Gaussian random matrix ensembles with Dyson indices β_D ∈ {1, 2, 4}, initialized in a coherent Gibbs state (τ = (β, 0)), beyond the known βΔ = 0 case.

Background

The authors examine a solvable two-dimensional random matrix model, compute the spread complexity for a single instance, and then average over Gaussian ensembles characterized by Dyson indices. They obtain integral representations for the ensemble averages and display numerical results for various β.

While closed-form expressions are known at βΔ = 0, the authors state that they do not know analytic expressions for generic β. Solving this would provide explicit temperature dependence of complexity in the simplest RMT setting and serve as a benchmark for understanding higher-dimensional ensembles.

References

Several questions remain open. Our analysis focused primarily on the infinite-temperature TFD state, for which the spectral weights are uniform. It would be interesting to determine how the picture changes at finite temperature or for more general initial states, where nontrivial spectral weights provide an additional source of structure.

— Unfolded Krylov complexity: universal chaotic dynamics without false positives  (2609.02228 - Erdmenger et al., 2 Sep 2026) in Section 7, Discussion