Systematic derivation of the conjugation relation between saddle-point parameters

Derive, in a rigorous and self-contained manner, the conjugation relation \(\tilde u = -\bar u\) for the saddle-point parameters in the large-\(k\) analysis of sparse non-Hermitian random matrices, starting from the general operator identity \(\tilde g = \operatorname{sign}(\hat C)\,\bar g\) where \(\hat C\) is defined by the kernel \(C(X,X') = k\,[1-\hat p\big(i\tilde X^{\dagger}X - iX^{\dagger}\tilde X\big)]\). Establish the conditions under which this relation holds and justify its use in obtaining the circular law and spectral-edge shapes for the ensembles considered.

Background

In the supersymmetry-based field-theoretic treatment, the averaged resolvent of sparse non-Hermitian ensembles reduces to a saddle-point equation for functions $g$ and $\tilde g$ that are assumed superrotationally invariant. In the large-$k$ limit, these functions are approximated by linear forms $g(X)=k\,u\,X^{\dagger}X$ and $\tilde g(X)=k\,\tilde u\,X^{\dagger}X$, leading to algebraic self-consistency conditions.

To complete the solution and recover the circular law (and more general spectral-edge shapes in later sections), the analysis assumes the conjugation relation $\tilde u = -\bar u$. The authors note that a systematic derivation of this relation from the operator identity $\tilde g = \operatorname{sign}(\hat C)\,\bar g$ is currently missing and is left for future work, as its justification would strengthen the theoretical foundation and general applicability of the method.