GEPP growth factor for general random butterfly matrices

Prove that for a scalar random butterfly matrix sampled from the uniform-angle model, the entrywise Gaussian-elimination-with-partial-pivoting growth factor is almost surely attained in the final upper-triangular factor U, so that rho(B) equals the maximum-entry norm of U divided by the maximum-entry norm of B.

Background

For simple scalar butterfly matrices, prior work establishes that the maximum entry growth occurs in the final U factor, yielding a closed-form growth-factor description. The paper explains that this argument does not directly extend to general butterfly matrices because elimination of the leading block acts on a trailing block generated by a different recursive factor. Numerical experiments support the analogous statement for general random butterfly matrices, but the authors leave it as a conjecture.

References

For B \sim B(N,\Sigma_S), the entrywise GEPP growth factor is almost surely attained somewhere in the final factor U, i.e., \rho(B) = |U|{\max}/|B|{\max}.

— Random permutations using GEPP  (2610.10481 - Gunawan et al., 7 Oct 2026) in Conjecture 2, Section 4.1, “Growth factors of random general butterfly matrices”

the corresponding analysis for random butterfly matrices remains open.

— Random permutations using GEPP  (2610.10481 - Gunawan et al., 7 Oct 2026) in Section 6, “Discussion and future directions”