Bulk universality for sparse real non-Hermitian random matrices away from the real axis

This work proves bulk universality for sparse real non-Hermitian random matrices with iid, possibly N-dependent, entries.The entries have zero mean, variance N⁻¹, and satisfy sparse moment bounds of the formE|xᵢⱼ|ʳ ≤ Cᵣ / (N qʳ⁻²),with q ≥ N^κ for some fixed κ > 0.At complex bulk points uniformly separated from both the spectral edge and the real axis, the microscopic k-point eigenvalue statistics converge to the Ginibre bulk limit. More precisely, for |z₀| ≤ 1 − δ and |Im z₀| ≥ c, the rescaled non-real eigenvalue correlation functions agree with the Ginibre bulk statistics up to a negative power of N. RM1_I_REINFORCED_2026-10-02The proof addresses two structural difficulties specific to sparse real non-Hermitian matrices.First, real-entry covariance creates transpose contractions. Under Hermitisation, transposition exchanges the spectral labels z and z̄, so the natural local-law class is not a one-label hierarchy but a binary family indexed by words in {z, z̄}. The paper proves averaged and isotropic multi-resolvent local laws for this entire binary class, preserving the same N-, η-, and sparsity-q powers as in the sparse complex theory. RM1_I_REINFORCED_2026-10-02Second, a non-real eigenvalue of a real matrix occurs together with its conjugate. A partial-Schur step therefore removes a real two-dimensional invariant subspace. After Hermitisation, this becomes a rank-four Feshbach correction. The associated weighted real O(n,2) Stiefel measure creates transpose pairings absent from the unitary setting, but these pairings remain inside the same binary z/z̄ resolvent hierarchy. RM1_I_REINFORCED_2026-10-02A central deterministic ingredient is the mixed z/z̄ stability operator. On the relevant test-matrix space, the potentially singular block satisfiesdet B_danger = 4(Im z)² + O(|w₁| + |w₂|).Hence the mixed stability inverse is uniformly bounded whenever the bulk point remains a fixed positive distance from the real axis. This identifies the distance to the real axis as the natural stability gap in the fixed-distance theory. RM1_I_REINFORCED_2026-10-02The paper then proves a sparse real binary multi-resolvent local law for arbitrary fixed binary words. Real entry differentiation, transpose operations, finite-N covariance remainders, exceptional cumulant configurations, and fluctuation averaging are shown to remain closed within the binary hierarchy without loss in the target N-, η-, or q-exponents. RM1_I_REINFORCED_2026-10-02These local laws are propagated through successive real codimension-two Schur compressions. The proof uses an exact rank-four Feshbach identity together with weighted-Stiefel concentration and a finite high-order saddle expansion. This replaces the low-order truncations that would otherwise impose stronger short-time restrictions. RM1_I_REINFORCED_2026-10-02The resulting short-time real Gauss-divisible theorem reaches Gaussian timest ≥ N⁻¹⁺²ε.At this scale, high-order expansions of determinant ratios, Pfaffian saddle factors, and weighted-Stiefel terms are controlled by the binary multi-resolvent law and the propagated projection estimates. The leading saddle is the same complex-bulk saddle as in the dense real theory, and the limiting local kernel is the Ginibre determinant kernel. RM1_I_REINFORCED_2026-10-02Finally, the Gaussian component is removed by an entrywise comparison. The first two cumulants match exactly, so the expansion begins at third order. The sparse moment structure provides the q⁻¹ gain needed to make the total replacement error negligible when the Gaussian time is chosen appropriately. RM1_I_REINFORCED_2026-10-02The paper develops the fixed-distance sparse-real universality engine away from the real axis. A companion work extends the same binary Hermitisation and partial-Schur architecture toward the real axis, down to polynomial distances |Im z_N| ≥ N⁻¹ᐟ²⁺ε, where the mixed stability gap becomes scale-dependent.