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.