LOCALIZATION FOR RANDOM QUASI-ONE-DIMENSIONAL DIRAC OPERATORS

The paper proves, for quasi-1D random Dirac models with potentials supported on Pauli matrices, a classification that matches the candidate’s claims: (i) in Case 1 the Furstenberg group is compact and all Lyapunov exponents vanish with purely a.c. spectrum; (ii) in Cases 2–4, for small cell size ℓ and energies in a suitable window (up to a finite exceptional set after a genericity step), one has G(E)=Sp_N(R), yielding localization; (iii) in Case 5 a time-reversal constrained subgroup arises, producing a parity effect with a single zero pair of Lyapunov exponents for N odd and localization for N even (Theorem 1.11) . The technical route also aligns: the one-step transfer is exp(ℓX), Lemma 3.6 provides a logarithm chart for small ℓ, Breuillard–Gelander (Theorem 3.5) is used to pass Lie-algebra generation to density, and an explicit commutator computation establishes the generated Lie algebras for Δ, followed by a genericity argument in V to obtain the finite exceptional set S_V (Proposition 3.7 and Section 3.3) . Minor differences are mostly nomenclature/packaging: the paper works with the subgroup SpO_N(R) in Case 5 and derives the parity phenomenon (Theorems 3.2–3.4) , whereas the candidate refers to SO*(2N). Both frameworks encode the same symmetry constraint and lead to the same spectral conclusions.