Lyapunov Exponents and Stability Properties of Higher Rank Representations

The paper proves, under the standing assumptions (each parameter yields a strongly irreducible and proximal representation), that χ1 and χd+1 are pluriharmonic if and only if the family is proximally stable, and that Tbif := ddc(χ1+χd+1) vanishes exactly on the stability locus with support equal to the bifurcation locus. These statements and their proofs appear explicitly: definitions and main theorem (Theorem 4) and the Tbif interpretation are stated in the results section, including “support equals bifurcation locus” . The “easy” direction (proximal stability ⇒ pluriharmonic exponents) uses random-product approximants and a Hartogs-type lemma so that, once v_n(λ) = (1/n)log r(ρ_λ(γ(n))) becomes pluriharmonic by proximal stability, the limit χ1 is pluriharmonic; χd+1 follows from duality . The converse (pluriharmonicity ⇒ proximal stability) reduces to one complex dimension, uses the superharmonic gap χ1−χ2 to propagate proximality, and then constructs a holomorphic variation of the Poisson boundary via control of the volume of random graph iterates by Tbif, ultimately forcing attracting/repelling data to move holomorphically and hence stability . The candidate solution recapitulates the same equivalence and proof strategy with essentially the same ingredients (psh approximants, Hartogs lemma, duality; area control of graphs; holomorphic variation of the Poisson boundary; non-intersection), differing mainly in the technical choice of approximants (integrated spectral-radius potentials vs. fixed typical word) and in a few unproved regularity estimates. These differences are minor and fixable; they do not alter the core logic. Hence, both are correct and substantially the same proof.