Quantitative Analyticity for Lyapunov Exponents of Random Products of Matrices with Explicit Polydiscs and Cauchy Coefficient Bounds

Theorem 1.1. (Quantitative analyticity in the weights). Let $A = (A_1, ..., A_N) \in GL(d, \mathbb{R})^N$ and $p^0 \in \Delta_N^{\circ}$ with $\lambda_+(A, p^0) > \lambda_2(A, p^0)$ (simple top Lyapunov exponent). Then there exist - (i) an explicit polydisc radius $r_*(A, p^0, \theta) > 0$ , - (ii) an explicit constant $M_*(A, p^0, \theta) < \infty$ , such that the Lyapunov exponent $p \mapsto \lambda_+(A,p)$ extends to a holomorphic function $\widetilde{\lambda}_+: D_{r_*(A,p^0,\theta)}(p^0) \to \mathbb{C}$ with (1.5) $$\sup_{z \in D_{r_(A,p^0,\theta)}(p^0)} |\widetilde{\lambda}_+(z)| \le M_(A,p^0,\theta).$$ The extension $\widetilde{\lambda}_+$ agrees with $\lambda_+(A,\cdot)$ on $D_{r_*}(p^0) \cap \mathbb{R}^N$ . Proof: See Section 4. Theorem 1.2: explicit Cauchy coefficient bounds.

Theorem 1.2. (Cauchy bounds for Taylor coefficients). Under the hypotheses of Theorem 1.1, for every multi-index $\alpha = (\alpha_1, \dots, \alpha_N) \in \mathbb{N}^N$ with $|\alpha| = \sum \alpha_i$ , the Taylor coefficients of $\lambda_+(A, p)$ at $p^0$ satisfy $$|\partial_p^{\alpha} \lambda_+(A, p^0)| \le \alpha! \cdot \frac{M_(A, p^0, \theta)}{r_(A, p^0, \theta)^{|\alpha|}},$$ where $r_(A, p^0, \theta)$ and $M_(A, p^0, \theta)$ are the constants from Theorem 1.1. Consequently, the Taylor series of $\lambda_+(A, p)$ at $p^0$ converges absolutely on the polydisc $D_{r_*(A, p^0, \theta)}(p^0)$ . Proof: See Section 5. Theorem 1.3: joint analyticity in weights and matrices.

Theorem 1.3. (Joint analyticity). Let $(A^0, p^0) \in GL(d, \mathbb{R})^N \times \Delta_N^\circ$ with $\lambda_+(A^0, p^0) > \lambda_2(A^0, p^0)$ . Then there exist explicit radii $r_^A(A^0, p^0, \theta) > 0$ and $r_^p(A^0, p^0, \theta) > 0$ such that the Lyapunov exponent $(A, p) \mapsto \lambda_+(A, p)$ extends to a holomorphic function on the product polydisc $$(1.7) D_{r^A}(A^0) \times D_{r^P}(p^0) \subset \mathbb{C}^{Nd^2} \times \mathbb{C}^N,$$ where $D_{r_^A}(A^0)$ is the polydisc in $\mathbb{C}^{Nd^2}$ around $A^0 = (A_1^0, \dots, A_N^0)$ whose components have radius $r_^A$ . Proof: See Section 6. Theorem 1.4: Markov chain extension.

Theorem 1.4. (Markov chain analyticity). Let $P = (P_{ij})_{i,j=1}^N$ be a stochastic matrix on $\{1,\ldots,N\}$ with all entries strictly positive, and let $A = (A_1,\ldots,A_N) \in \operatorname{GL}(d,\mathbb{R})^N$ . Let $\lambda_+(P,A)$ denote the top Lyapunov exponent of the Markov cocycle with transition matrix P and fiber matrices A. Assume $\lambda_+(P,A) > \lambda_2(P,A)$ . Then there exist explicit radii $r_^P(P, A, \theta) > 0$ and $r_^A(P, A, \theta) > 0$ such that $(P, A) \mapsto \lambda_+(P, A)$ extends to a holomorphic function on the product polydisc $$(1.8) D_{r_{\cdot}^{P}}(P) \times D_{r_{\cdot}^{A}}(A) \subset \mathbb{C}^{N^{2}} \times \mathbb{C}^{Nd^{2}}.$$ The radius $r_*^P$ depends on the spectral gap $\rho_P$ of P. Proof: See Section 7. Theorem 1.5: boundary behavior.

Theorem 1.5. (Boundary behavior near the simplex boundary, conditional). Let $A = (A_1, \ldots, A_N) \in GL(d, \mathbb{R})^N$ and $p^0 \in \Delta_N^{\circ}$ with $\lambda_+(A, p^0) > \lambda_2(A, p^0)$ . Fix a coordinate index $j \in \{1, \ldots, N\}$ and consider the one-parameter family $p(t) = p^0 - te_j + (t/(N-1)) \sum_{i \neq j} e_i$ (moving $p^0$ toward the boundary $p_j = 0$ ). Assume that A satisfies Hypothesis 8.1 (polynomial spectral-gap decay) with constants $c_{\tau}, \gamma_{\tau}$ . Then as $t \to p_j^0$ , the radius of analyticity $r_*(A, p(t), \theta)$ decays no faster than a power of $(p_j^0 - t)$ : $$(1.9) r_*(A, p(t), \theta) \ge c_E \cdot (p_i^0 - t)^{\alpha_E}$$ for some explicit constants $c_E, \alpha_E > 0$ depending only on A and $\theta$ . This quantifies the controlled degeneration of the analytic extension as the weights approach the simplex boundary, in those cases where the spectral gap degrades polynomially (cf. Remark 8.2). Proof: See Section 8. Theorem 1.6: quantitative polydisc for GL(d).

Theorem 1.6. (GL(d) extension). The conclusions of Theorems 1.1 through 1.4 hold for GL(d, $\mathbb{R}$ ) for every $d \geq 2$ , with the same closed-form constants computed from the eccentricity of the matrices, the spectral gap of the Markov operator on $\mathbb{P}^{d-1}$ , and the Lyapunov gap $\lambda_1 - \lambda_2$ (in place of the GL(2) quantity $\lambda_+ - \lambda_-$ ). The extension uses the Fubini-Study metric on projective space and the action of $\Lambda^2 g$ on bivectors. Proof: See Section 9. Theorem 1.7: sub-top Lyapunov exponents under strong k-irreducibility.

Theorem 1.7. (Quantitative analyticity of the partial sums in $GL(d, \mathbb{R})$ ). Let $d \geq 2$ , $1 \leq k \leq d-1$ , and $\theta \in (0,1]$ . Let $A^0 \in GL(d, \mathbb{R})^N$ be strongly k-irreducible with simplicity gap $\lambda_k(A^0, p^0) > \lambda_{k+1}(A^0, p^0)$ at $p^0 \in \Delta_N^\circ$ . Then there exists an explicit polydisc radius $r_H^{(k)}(A^0, p^0, \theta) > 0$ such that the partial sum $\Lambda_k(A, p) = \lambda_1(A, p) + \cdots + \lambda_k(A, p)$ extends to a holomorphic function on the polydisc of radius $r_H^{(k)}$ in both the weight vector $p \in \mathbb{C}^N$ and the matrix coefficients $A \in GL(d, \mathbb{C})^N$ . The proof and the explicit formula for $r_H^{(k)}$ are given in Section 10 (Theorem 10.8). A corollary (Corollary [10.9\)](#page-33-0) yields the analogous analyticity of the individual subtop Lyapunov exponents λk(A, p) by subtracting consecutive partial sums. <span id="page-7-0"></span>1.4. Relation to prior work. This subsection situates the paper within the literature on Lyapunov exponents of random matrix products, in three layers: the foundational continuity theory, the analyticity theory of Peres and its extensions, and the quantitative direction in which we work. Continuity of Lyapunov exponents: classical theory. The study of Lyapunov exponents of random matrix products begins with the seminal works of [Furstenberg](#page-47-1) [\(1963\)](#page-47-1); [Furstenberg and Kesten](#page-47-0) [\(1960\)](#page-47-0), who established the existence and almostsure convergence of Lyapunov exponents for i.i.d. products under integrability hypotheses. Continuity of the top Lyapunov exponent in the probability distribution under irreducibility hypotheses was proved by [Furstenberg and Kifer](#page-47-2) [\(1983\)](#page-47-2) and refined by [Hennion](#page-47-3) [\(1984\)](#page-47-3); [Le Page](#page-48-2) [\(1982\)](#page-48-2) and [Guivarc'h and Raugi](#page-47-4) [\(1985\)](#page-47-4) independently developed Kato perturbation arguments for the projective Markov operator that established H¨older continuity in restricted settings. [Goldsheid and](#page-47-17) [Margulis](#page-47-17) [\(1989\)](#page-47-17) gave a definitive treatment of Lyapunov spectrum simplicity for random matrix products and clarified the role of the Zariski closure of the support. In the past decade, qualitative continuity has been substantially extended. [Bocker](#page-47-6) [and Viana](#page-47-6) [\(2017\)](#page-47-6) proved that the top Lyapunov exponent is continuous at every compactly supported probability measure on GL(2, R), without any irreducibility assumption. [Malheiro and Viana](#page-48-3) [\(2015\)](#page-48-3) extended this continuity result to Markov chain driven cocycles in GL(2). [Backes, Brown, and Butler](#page-46-3) [\(2018\)](#page-46-3) extended continuity to a broad class of linear cocycles with invariant holonomies. [Avila, Eskin,](#page-46-0) [and Viana](#page-46-0) [\(2023\)](#page-46-0) established the continuity of Lyapunov exponents in GL(d) for arbitrary dimension d ≥ 2. [Poletti and Viana](#page-48-7) [\(2019\)](#page-48-7) gave criteria for the simplicity of the Lyapunov spectrum in the closely related setting of partially hyperbolic linear cocycles. For non-compactly supported measures, [Sanchez and Viana](#page-48-8) [\(2019\)](#page-48-8) proved that the Lyapunov exponents are semi-continuous with respect to the Wasserstein topology in GL(2) but not with respect to the weak-∗ topology, identifying moment conditions as essential. A modern reference for the continuity theory in the broader setting of linear cocycles (with detailed treatment of the projective and Grassmannian methods) is [Arnold](#page-46-4) [\(1998\)](#page-46-4); [Viana](#page-48-9) [\(2014\)](#page-48-9); [Duarte and Klein](#page-47-7) [\(2016\)](#page-47-7). The companion paper [Thiam](#page-48-0) [\(Nov. 2025\)](#page-48-0) establishes quantitative H¨older and log-H¨older moduli of continuity for the present setting, with explicit constants. Analyticity: from Peres to Bezerra-Sanchez-Tall. Beyond continuity, the question of analytic dependence on the data was first addressed by [Ruelle](#page-48-6) [\(1979\)](#page-48-6), who proved analyticity of Lyapunov exponents for products of random positive matrices. The general result is due to [Peres](#page-48-5) [\(1991\)](#page-48-5), who proved that for a finitely supported i.i.d. random matrix product, the top Lyapunov exponent depends real-analytically on the probability weights p, provided the top Lyapunov exponent is simple. The proof uses a Kato perturbation argument applied to the transfer operator on projective space. The Peres theorem has been extended in two directions. [Bezerra, Sanchez,](#page-47-10) [and Tall](#page-47-10) [\(2021\)](#page-47-10) proved real-analyticity of the top Lyapunov exponent in the probability weights for a random product of quasi-periodic cocycles, where the matrices Aⁱ = Ai(t) depend continuously on a torus variable t ∈ T ^m. [Baraviera and Duarte](#page-46-2) [\(2025\)](#page-46-2) extended the Peres theorem in an orthogonal direction: they considered a compact but possibly infinite symbol space, and proved analyticity of the top Lyapunov exponent with respect to the total variation norm, using complex analysis in Banach spaces. The extension of [Baraviera and Duarte](#page-46-2) [\(2025\)](#page-46-2) is complementary to ours: they enlarge the symbol space from finite to compact-infinite at the price of working in the total variation topology, while we keep the finite symbol space of Peres and provide explicit polydiscs and Cauchy bounds for the Taylor coefficients. In a related direction, [Avila](#page-46-5) [\(2008\)](#page-46-5); [Avila and Jitomirskaya](#page-46-6) [\(2009\)](#page-46-6) prove H¨older continuity of the Lyapunov exponent of one-frequency Schr¨odinger cocycles in the rotation number under Diophantine conditions, in the analytic regime. Their setting complements ours by fixing the random weights and varying the base dynamics; the H¨older modulus they obtain in the rotation number is, in general, not improvable to analyticity (discussed in Subsection [13.4\)](#page-44-0). Quantitative direction: this paper and parallel work. The qualitative analyticity theorems of [Peres](#page-48-5) [\(1991\)](#page-48-5) and [Bezerra, Sanchez, and Tall](#page-47-10) [\(2021\)](#page-47-10) establish the existence of a real-analytic extension on a small neighborhood of the base point, but provide neither the radius of the neighborhood nor any quantitative information about the Taylor coefficients. The principal aim of this paper is to convert these qualitative statements into quantitative ones: explicit polydiscs of holomorphy in C ^N , explicit Cauchy bounds for the Taylor coefficients, joint analyticity in the matrices, Markov chain extension, boundary behavior, and a GL(d, R) extension at the level of the top exponent and (under strong k-irreducibility) of all sub-top partial sums. This work is parallel to several recent quantitative developments. [Duarte and](#page-47-8) [Klein](#page-47-8) [\(2019\)](#page-47-8) obtained quantitative large deviation estimates for iterates of linear cocycles. [Duarte, Klein, and Santos](#page-47-9) [\(2020\)](#page-47-9) constructed an explicit example of a random matrix cocycle whose Lyapunov exponent is not H¨older continuous in the weight, establishing a sharp obstruction to the regularity that any continuity theorem can hope to achieve. [Baraviera and Duarte](#page-46-1) [\(2019\)](#page-46-1) gave a constructive proof of the Le Page H¨older continuity theorem for irreducible Bernoulli cocycles, with an algorithm to approximate the Lyapunov exponent and the stationary measure. [Bezerra and Duarte](#page-47-15) [\(2022\)](#page-47-15) proved upper bounds on the regularity of the Lyapunov exponent that complement the lower bounds (such as our Proposition [12.4\)](#page-40-0). [Kogler](#page-47-13) [\(2020\)](#page-47-13) derived quantitative lower bounds for the positivity of the Lyapunov exponent via Golden-Thompson inequalities and the avalanche principle, and [Bedrossian and Wu](#page-47-14) [\(2024\)](#page-47-14) proved a quantitative dichotomy in the small noise limit for non-dissipative SDEs. These results quantify the positivity of λ+(A, p), while our Theorems [1.1,](#page-5-0) [1.2,](#page-5-1) [1.3,](#page-5-2) [1.4,](#page-6-1) [1.5,](#page-6-2) [1.6,](#page-6-3) and [1.7](#page-6-0) quantify the regularity of the map (A, p) 7→ λ+(A, p). Finally, [Graxinha](#page-47-16) [\(2025\)](#page-47-16) showed that finite moment conditions are essential for the modulus of continuity in the non-compact case; our work is confined to the compactly supported case. The Kato perturbation machinery itself, on which our entire approach rests, is a standard tool from operator theory [Kato](#page-47-18) [\(1980\)](#page-47-18); its application to random matrix products in the context of central limit theorems goes back to [Le Page](#page-48-2) [\(1982\)](#page-48-2); [Guivarc'h and Raugi](#page-47-4) [\(1985\)](#page-47-4). 1.5. Organization. Section [2](#page-9-0) fixes notation and defines the Markov operator. Section [3](#page-11-0) sets up the complex Markov operator and its spectral theory. Section [4](#page-14-0) proves the quantitative polydisc of Theorem [1.1.](#page-5-0) Section [5](#page-18-0) proves the Cauchy coefficient bounds of Theorem 1.2. Section 6 proves Theorem 1.3 on joint analyticity. Section 7 proves Theorem 1.4 on Markov chain extension. Section 8 proves Theorem 1.5 on boundary behavior. Section 9 extends to $\mathrm{GL}(d,\mathbb{R})$ and proves Theorem 1.6. Section 10 proves Theorem 10.8 on the sub-top Lyapunov exponents under strong k-irreducibility. Section 11 computes numerical values for a concrete example. Section 12 contains the method-optimality proposition for the Kato polydisc, the structural-obstruction theorem identifying the complex spectral collapse set, and the Bernstein-type sharpness of the Cauchy bounds. Section 13 gathers open problems. Appendices collect technical lemmas.

Lemma 2.1. (Projective contraction). For every $g \in GL(d, \mathbb{R})$ and every $[u], [v] \in \mathbb{P}^{d-1}$ . (2.5) $$d(g[u], g[v]) \le ecc(g)^2 \cdot d([u], [v]),$$ where $ecc(g) = ||g|| \cdot ||g^{-1}||$ .

Lemma 2.2. (Log-norm Lipschitz bounds). Set $\phi(g, [v]) = \log(\|gv\|/\|v\|)$ . For every $g, g' \in GL(d, \mathbb{R})$ and $[u], [v] \in \mathbb{P}^{d-1}$ , $$(2.6) |\phi(g, [v]) - \phi(g', [v])| \le \max(\|g^{-1}\|, \|g'^{-1}\|) \cdot \|g - g'\|,$$ $$(2.7) |\phi(q, [u]) - \phi(q, [v])| < (||q|| \cdot ||q^{-1}|| + 1) \cdot d([u], [v]).$$ 2.4. Spectral gap of the Markov operator. The central technical fact on which the analyticity theory depends is the spectral gap of the Markov operator on the Hölder space. We record the relevant statement, whose proof is given in Thiam (Nov. 2025) (quantitative version) and in (Bocker and Viana, 2017, Section 5) (qualitative version).

Proposition 2.3. (Spectral gap on $C^{\theta}$ ). Let $A \in GL(d, \mathbb{R})^N$ and $p \in \Delta_N^{\circ}$ with simple top Lyapunov exponent $\lambda_+(A, p) > \lambda_2(A, p)$ . Then for every $\theta \in (0, \theta_(A, p))$ , where $\theta_(A, p) > 0$ is an explicit threshold, there exist $N_{\theta} \geq 1$ and $\tau(A, p, \theta) \in (0, 1)$ such that $$(2.8)$$ Explicit formulas for $\theta_*(A, p)$ , $N_{\theta}$ , and $\tau(A, p, \theta)$ are given in (Thiam, Nov. 2025, Theorem 3.3). Proof: See (Thiam, Nov. 2025, Section 4); we use this result as input to the complex-perturbation arguments below. For the analyticity theorems of this paper, the key quantity is the spectral gap magnitude $1-\tau$ , which determines the radius of analyticity. We shall often write $\tau_*$ for $\tau(A, p, \theta)$ when the dependence is understood. 2.5. Furstenberg-Khasminskii formula. The Furstenberg-Khasminskii formula expresses the Lyapunov exponent as an integral against the stationary measure: <span id="page-11-4"></span>(2.9) $$\lambda_{+}(A,p) = \sum_{i=1}^{N} p_{i} \int_{\mathbb{P}^{d-1}} \phi(A_{i},[v]) d\eta_{A,p}^{+}([v]).$$ This identity is the starting point of all perturbation arguments in this paper. The stationary measure $\eta_{A,p}^+$ depends nontrivially on (A,p), and the task of the analyticity theory is to understand this dependence as a holomorphic function of complex p. 2.6. Complex Markov operators. For any $z = (z_1, \ldots, z_N) \in \mathbb{C}^N$ satisfying the normalization $\sum_i z_i = 1$ , we define the complex Markov operator (2.10) $$(P_{A,z}\varphi)([v]) = \sum_{i=1}^{N} z_i \, \varphi(A_i \cdot [v]).$$ Unlike the real case, the complex $z_i$ are not probabilities, but the operator $P_{A,z}$ is still a bounded linear map on $C^{\theta}(\mathbb{P}^{d-1})$ . When $z = p \in \Delta_N^{\circ}$ , we recover the real Markov operator of (2.2). The holomorphic extension of $\lambda_+$ that we construct will be expressed, via the Furstenberg-Khasminskii formula, in terms of the analytic perturbation of the leading eigenvalue of $P_{A,z}$ as a function of z.

Lemma 3.1. (Operator norm Lipschitz bound). Let $A \in GL(d, \mathbb{R})^N$ . For every $p, q \in \mathbb{C}^N$ , <span id="page-11-2"></span>(3.1) $$||P_{A,p} - P_{A,q}||_{C^{\theta} \to C^{\theta}} \le \sum_{i=1}^{N} |p_i - q_i| \cdot ||T_i||_{C^{\theta} \to C^{\theta}},$$ where $T_i\varphi([v]) = \varphi(A_i \cdot [v])$ is the individual transfer operator associated with $A_i$ . Moreover, <span id="page-11-3"></span> $$||T_i||_{C^\theta \to C^\theta} < 1 + \operatorname{ecc}(A_i)^{2\theta}.$$

Lemma 3.2. (Spectral gap of the real operator). The operator $P_{A,p^0}: C^{\theta} \to C^{\theta}$ - (i) A simple eigenvalue 1 with eigenspace $sp\{1\}$ and dual eigenspace $sp\{\eta^+\}$ . - (ii) All other spectrum contained in the closed disc {ζ ∈ ℂ : |ζ| ≤ τ\_\^1/Nθ}. (iii) An isolating circle Γ\ ⊂ ℂ centered at ζ = 1 with radius ρ\ = (1 − τ\_\^1/Nθ)/2, enclosing only the eigenvalue 1. - 3.3. Complex perturbation. For the complex Markov operator $P_{A,z}$ with z in a neighborhood of $p^0$ , we want to transfer the spectral decomposition (3.3) using Kato's perturbation theory. <span id="page-12-3"></span>Lemma 3.3 (Resolvent bound on the isolating circle). There exists an explicit constant $K_(A, p^0, \theta) < \infty$ such that, for every $\zeta \in \Gamma$ (the isolating circle of Lemma 3.2), (3.4) $$\|(\zeta \operatorname{Id} - P_{A,p^0})^{-1}\|_{C^{\theta} \to C^{\theta}} \le K_*(A, p^0, \theta).$$ The constant has the explicit form $$(3.5) K_(A, p^0, \theta) = \frac{1}{\rho} + \frac{N_{\theta} \cdot ||R_{A, p^0}||_{C_0^{\theta} \to C_0^{\theta}}^{N_{\theta} - 1}}{(1 - \rho_)^{N_{\theta}} - \tau}, \rho_ := \frac{1 - \tau_^{1/N_{\theta}}}{2}.$$

Proposition 3.5. (Holomorphic eigenvalue perturbation). Let $A \in GL(d, \mathbb{R})^N$ , $p^0 \in \Delta_N^{\circ}$ with simple top Lyapunov exponent, and $\theta \in (0, \theta_*(A, p^0))$ . Set <span id="page-13-2"></span>(3.7) $$r_(A, p^0, \theta) = \frac{1}{4NK_(A, p^0, \theta) \max_i (1 + \operatorname{ecc}(A_i)^{2\theta})}.$$ Then for every $z \in \mathbb{C}^N$ with $\sum_i z_i = 1$ and $|z_i - p_i^0| < r_*$ for all i: - (i) The complex Markov operator $P_{A,z}$ has a unique simple eigenvalue $\mu(z)$ in the disc $|\zeta 1| < \rho_*$ , with all other spectrum outside this disc. - (ii) The eigenvalue $\mu: D_{r_*}(p^0) \cap \{\sum z_i = 1\} \to \mathbb{C}$ is a holomorphic function of z. - (iii) $\mu(p^0) = 1$ .

Lemma 4.1. (Lyapunov exponent as logarithmic derivative). Let $A \in GL(d, \mathbb{R})^N$ and $p \in \Delta_N^{\circ}$ with simple top Lyapunov exponent. Define, for $s \in \mathbb{R}$ small, the twisted Markov operator (4.1) $$(L_{A,p,s}\varphi)([v]) = \sum_{i=1}^{N} p_i e^{s\phi(A_i,[v])} \varphi(A_i \cdot [v]),$$ where $\phi(g, [v]) = \log(\|gv\|/\|v\|)$ . Let $\mu_s$ denote the leading eigenvalue of $L_{A,p,s}$ on $C^{\theta}(\mathbb{P}^{d-1})$ . Then (4.2) $$\lambda_{+}(A,p) = \frac{d}{ds} \log \mu_{s} \bigg|_{s=0}.$$

Lemma 4.2. (Holomorphic stationary measure). Under the hypotheses of Proposition 3.5, there exists a family of complex linear functionals $\eta_z \in (C^{\theta}(\mathbb{P}^{d-1}))$ (the dual space), indexed by $z \in D_{r}(p^0) \cap \{\sum z_i = 1\}$ , depending holomorphically on z, such that: - (i) $\eta_z \mathbf{1} = 1$ for all z. - (ii) $P_{A,z}^ \eta_z = \mu(z) \eta_z$ for all z, where $P_{A,z}$ is the adjoint on $(C^{\theta})^*$ . - (iii) At $z = p^0$ , $\eta_{p^0}$ is the integration against $\eta_{A,p^0}^+$ , i.e., $\eta_{p^0}(\varphi) = \int \varphi \, d\eta_{A,p^0}^+$ , and $\mu(p^0) = 1$ .

Proposition 4.3. (Analytic extension of $\lambda_+$ ). Under the hypotheses of Proposition 3.5, the function $p \mapsto \lambda_+(A,p)$ on $\Delta_N^{\circ} \cap D_{r_/2}(p^0)$ extends to a holomorphic function $\widetilde{\lambda}_+ : D_{r_/2}(p^0) \cap \{\sum z_i = 1\} \to \mathbb{C}$ defined by <span id="page-16-1"></span>(4.4) $$\widetilde{\lambda}_{+}(z) = \sum_{i=1}^{N} z_{i} \cdot \eta_{z}(\phi(A_{i}, \cdot)).$$

Proposition 5.1. (Cauchy integral formula on polydiscs). Let $f: D_R(z^0) \to \mathbb{C}$ be a holomorphic function on a polydisc $D_R(z^0) = \{z \in \mathbb{C}^N : |z_i - z_i^0| < R\}$ . For every multi-index $\alpha \in \mathbb{N}^N$ and every r < R, <span id="page-18-2"></span> $$\partial_z^{\alpha} f(z^0) = \frac{\alpha!}{(2\pi i)^N} \oint_{|\zeta_1 - z_1^0| = r} \cdots \oint_{|\zeta_N - z_N^0| = r} \frac{f(\zeta)}{(\zeta_1 - z_1^0)^{\alpha_1 + 1} \cdots (\zeta_N - z_N^0)^{\alpha_N + 1}} \, d\zeta_1 \cdots d\zeta_N.$$ Consequently, <span id="page-18-3"></span>(5.2) $$|\partial_z^{\alpha} f(z^0)| \le \frac{\alpha! \cdot \sup_{\zeta \in D_r(z^0)} |f(\zeta)|}{r^{|\alpha|}}.$$

Lemma 6.1. (Joint operator norm Lipschitz bound). Let $(A, p), (A', p') \in \mathbb{C}^{Nd^2} \times \mathbb{C}^N$ with all matrices invertible. Then (6.1) $$||P_{A,p} - P_{A',p'}||_{C^{\theta} \to C^{\theta}} \le L_A \cdot \max_{i} ||A_i - A'_i|| + L_p \cdot \sum_{i} |p_i - p'_i|,$$ where $L_A, L_p$ are explicit constants. Specifically, (6.2) $$L_p = \max_{i} (1 + \operatorname{ecc}(A_i^0)^{2\theta} + \rho_0^A),$$ (6.3) $$L_A = \max_i p_i \cdot K_{\text{mat}}(A^0, \rho_0^A, \theta),$$ where $K_{\text{mat}}$ is the matrix-Lipschitz constant of the transfer operator $T_i$ , bounded by an explicit function of $\max_i \|A_i^0\|, \|(A_i^0)^{-1}\|$ , and the perturbation radius $\rho_0^A$ . Proof. Writing $$P_{A,p} - P_{A',p'} = (P_{A,p} - P_{A,p'}) + (P_{A,p'} - P_{A',p'})$$ : The first difference is linear in $p - p'$ , bounded as in Lemma 3.1 by $L_p \cdot \sum_i |p_i - p'_i|$ . The second difference involves varying A at fixed p': $$(P_{A,p'} - P_{A',p'})\varphi([v]) = \sum_{i} p'_{i} \left[\varphi(A_{i}[v]) - \varphi(A'_{i}[v])\right].$$ The $C^{\theta}$ norm of this difference is controlled by the Hölder modulus of $\varphi$ times the projective-distance perturbation $d(A_i[v], A_i'[v])$ . By Lemma 2.1 (extended to $g \mapsto g$ Lipschitz), $$d(A_i[v], A_i'[v]) \le C_{\text{geom}}(A_i^0, \rho_0^A) \cdot ||A_i - A_i'||,$$ for some explicit constant $C_{\mathrm{geom}}$ depending on $A_i^0$ and the perturbation radius $\rho_0^A$ . Hence $[P_{A,p'}-P_{A',p'}]_{\theta} \leq K_{\mathrm{mat}} \cdot \max_i \|A_i-A'_i\|$ for a constant $K_{\mathrm{mat}}$ that can be made explicit in terms of the Hölder data. The sup-norm bound is similar. $\square$ 6.3. Joint polydisc radius. This subsection runs the Kato perturbation argument with both A and p varying, producing a holomorphic extension of the leading eigenvalue and spectral projection of $P_{A,z}$ on a product polydisc in (A,z). The result is Proposition 6.2, the joint analog of Proposition 3.5 from Section 3; it is the central input to the proof of Theorem 1.3 in the next subsection. <span id="page-20-1"></span>Proposition 6.2 (Joint holomorphic extension). There exist explicit radii (6.4) $$r_*^A(A^0, p^0, \theta) > 0,$$ (6.5) $$r_*^p(A^0, p^0, \theta) > 0$$ , such that for every $A \in \mathbb{C}^{Nd^2}$ with $||A_i - A_i^0|| < r_^A$ for all i, and every $z \in \mathbb{C}^N$ with $\sum z_i = 1$ and $|z_i - p_i^0| < r_^p$ : (i) The complex Markov operator $P_{A,z}$ has a unique simple eigenvalue $\mu(A,z)$ in the disc $|\zeta - 1| < \rho_*$ . (ii) $\mu(A,z)$ and the spectral projection $\Pi_{A,z}$ are jointly holomorphic in (A,z). (iii) $$\mu(A^0, p^0) = 1$$ . Proof. Choose (6.6) $$r_^A = \frac{1}{8L_AK}, \qquad r_^p = \frac{1}{8L_pK_N}.$$ Then, by Lemma 6.1, for (A, z) in the product polydisc, $$||P_{A,z} - P_{A^0,p^0}||_{op} \le L_A \cdot r_^A + L_p \cdot N \cdot r_^p \le \frac{1}{8K} + \frac{1}{8K} = \frac{1}{4K_*}.$$ Following the argument of Proposition 3.5, the resolvent is expressed as a Neumann series converging uniformly on the isolating circle $\Gamma_*$ , with coefficients that are joint polynomials in (A, z). Each polynomial is trivially holomorphic in (A, z), so the spectral projection $\Pi_{A,z}$ and the eigenvalue $\mu(A,z)$ are jointly holomorphic. $\square$ 6.4. Proof of Theorem 1.3. This subsection deduces Theorem 1.3 (joint real-analyticity of the Lyapunov exponent in (A, p)) from the joint holomorphic extension of Proposition 6.2. The argument is parallel to the proof of Theorem 1.1 in Section 4, with the projective Markov operator's z-only Kato perturbation replaced by the joint (A, z) Kato perturbation of Proposition 6.2. Proof of Theorem 1.3. By Proposition 6.2, the spectral projection and eigenvalue are jointly holomorphic in (A, z) on the product polydisc $D_{r_^A}(A^0) \times D_{r_^P}(p^0) \cap \{\sum z_i = 1\}.$ The joint holomorphic stationary functional $\eta_{A,z}$ and its evaluations on $\phi(A_i,\cdot)$ are jointly holomorphic by the same argument. Explicitly, the analog of (4.4) is (6.7) $$\widetilde{\lambda}_{+}(A,z) = \sum_{i=1}^{N} z_{i} \cdot \eta_{A,z}(\phi(A_{i},\cdot)).$$ Holomorphy: $z_i$ is entire, $\phi(A_i,\cdot) \in C^{\theta}(\mathbb{P}^{d-1})$ depends holomorphically on $A_i$ (since $\phi(g,[v]) = \log \|gv\| - \log \|v\|$ , which is holomorphic in g when g is non-degenerate), and $\eta_{A,z}$ depends jointly holomorphically on (A,z) by Proposition 6.2. Hence $\widetilde{\lambda}_+(A,z)$ is jointly holomorphic. On the real domain $(A,p) \in \mathbb{R}^{Nd^2} \times \Delta_N^{\circ}$ , it agrees with $\lambda_+(A,p)$ by the Furstenberg-Khasminskii formula. This proves Theorem 1.3. Remark 6.3 (Joint Cauchy bounds). Applying the multi-dimensional Cauchy formula of Proposition 5.1 to the joint holomorphic extension, we obtain for every multi-indices $\alpha \in \mathbb{N}^{Nd^2}$ (for A) and $\beta \in \mathbb{N}^N$ (for p): $$(6.8) |\partial_A^{\alpha} \partial_p^{\beta} \lambda_+(A^0, p^0)| \le \alpha! \beta! \cdot \frac{M_^{\text{joint}}}{(r_^A)^{|\alpha|}(r_*^p)^{|\beta|}},$$ where $M_*^{\mathrm{joint}}$ is the supremum of $|\widetilde{\lambda}_+|$ on the product polydisc. This joint bound yields, as a special case at $\alpha = 0$ , the weight-only Cauchy bound of Theorem 1.2. At $\beta = 0$ , it gives a new bound on the matrix derivatives of $\lambda_+$ , which is of independent interest in sensitivity analysis.

Proposition 7.1. (Spectral gap of the Markov-chain operator). Let P be strictly positive stochastic with spectral gap $\rho_P \in (0,1)$ , and $A \in GL(d,\mathbb{R})^N$ with $\lambda_+(P,A) > \lambda_2(P,A)$ . For every $\theta \in (0,\theta_*(P,A))$ , there exist an integer $N_{\theta}(P,A) \geq 1$ and a spectral gap $\tau(P,A,\theta) < 1$ such that (7.4) $$||Q_{P,A}^{N_{\theta}}\varphi||_{C^{\theta}} \leq \tau(P,A,\theta) \cdot ||\varphi||_{C^{\theta}}, \qquad \varphi \in C_0^{\theta}(\{1,\ldots,N\} \times \mathbb{P}^{d-1}).$$ The constant $\tau(P, A, \theta) \leq \max(\rho_P, \tau_0(A, \theta))^c$ for some explicit exponent $c \in (0, 1)$ depending on $\theta$ and the dimensions of P and A.

Lemma 7.2. (Operator norm Lipschitz in (P,A)). For (P,A), (P',A') with all transition-matrix entries within distance $\rho_0^P$ of the base entries, $$(7.5) ||Q_{P,A} - Q_{P',A'}||_{C^{\theta} \to C^{\theta}} \le L^{P} \cdot \max_{i,j} |P_{ij} - P'_{ij}| + L^{A} \cdot \max_{i} ||A_{i} - A'_{i}||,$$ with explicit constants $L^P, L^A$ : $$(7.6) L^P = N \cdot \max_i ||T_i||_{C^\theta},$$ (7.6) $$L^{P} = N \cdot \max_{i} ||T_{i}||_{C^{\theta}},$$ (7.7) $$L^{A} = K_{\text{mat}}^{\text{chain}}(P, A, \theta, \rho_{0}^{A}),$$ where $||T_i||_{C^{\theta}} \leq 1 + \mathrm{ecc}(A_i)^{2\theta}$ and $K_{\mathrm{mat}}^{\mathrm{chain}}$ is the matrix-Lipschitz constant for the Markov-chain setup. Proof. Decompose $Q_{P,A} - Q_{P',A'} = (Q_{P,A} - Q_{P',A}) + (Q_{P',A} - Q_{P',A'})$ . The first difference involves varying only P at fixed A: by linearity of Q in $P_{ij}$ , $$(Q_{P,A} - Q_{P',A})\varphi(i,[v]) = \sum_{j} (P_{ij} - P'_{ij})\varphi(j,A_{j}[v]).$$ The $C^{\theta}$ norm of this is at most $N \cdot \max_{i} ||T_{i}||_{C^{\theta}} \cdot \max_{i,j} |P_{ij} - P'_{ij}|$ , giving the $L^{P}$ bound. The second difference involves varying only A at fixed P: the argument is parallel to Lemma 6.1, giving the $L^A$ bound. 7.5. Proof of Theorem 1.4. This subsection deduces Theorem 1.4 (joint analyticity in the transition matrix and the cocycle for Markov-chain driven cocycles) by applying the Kato perturbation scheme of Section 3 to the extended operator $Q_{P,A}$ on $C^{\theta}(\{1,\ldots,N\}\times\mathbb{P}^{d-1})$ instead of the projective Markov operator. The spectral gap input is provided by Proposition 7.1; the Lipschitz input by Lemma 7.2. Proof of Theorem 1.4. Applying the Kato perturbation argument (as in Section 3) to the operator $Q_{P,A}$ on $C^{\theta}(\{1,\ldots,N\}\times\mathbb{P}^{d-1})$ , with the spectral gap of Proposition 7.1 and the Lipschitz bound of Lemma 7.2, we obtain: The eigenvalue $\mu(P,A)$ of $Q_{P,A}$ near 1 is jointly holomorphic in (P,A) on the product polydisc $$(7.8) |P_{ij} - P_{ij}^0| < r_^P := \frac{1}{8L^P K_^{\text{chain}}}, ||A_i - A_i^0|| < r_^A := \frac{1}{8L^A K_^{\text{chain}}},$$ where $K_*^{\text{chain}} = 4/(1-\tau(P,A,\theta)^{1/N_\theta})$ is the resolvent bound in the Markov-chain The Furstenberg-Khasminskii formula for the Markov-chain Lyapunov exponent is (7.9) $$\lambda_{+}(P,A) = \sum_{i,j} \pi_{i}(P) P_{ij} \int \phi(A_{j},[v]) d\eta_{P,A}^{+}(i,[v]),$$ where $\eta_{P,A}^+$ is the unique stationary measure on $\{1,\ldots,N\}\times\mathbb{P}^{d-1}$ for $Q_{P,A}$ , and $\pi(P)$ is the stationary distribution of the chain (a rational function of P, hence holomorphic). The holomorphic extension of $\eta_{P,A}^+$ (as a complex linear functional on $C^{\theta}$ ) is obtained from the spectral projection $\Pi_{P,A}^{\text{chain}}$ as in Lemma 4.2. Combining all pieces, the extension (7.10) $$\widetilde{\lambda}_{+}(P,A) = \sum_{i,j} \pi_{i}(P) P_{ij} \, \eta_{P,A}^{+}(\phi(A_{j},\cdot) \cdot \mathbf{1}_{j})$$ is jointly holomorphic on the product polydisc. On the real domain, it agrees with the Markov-chain Lyapunov exponent $\lambda_{+}(P, A)$ . The dependence of $r_^P$ on $\rho_P$ is through the factor $\tau(P,A,\theta)$ , which degrades as $\rho_P \to 0$ (weak base-chain mixing). The explicit formula from Proposition 7.1 makes this dependence quantitative: $1 - \tau \sim (1 - \rho_P)^{c'}$ for some explicit exponent c', so $r_^P \sim (1 - \rho_P)^{c'}$ as $\rho_P \to 1$ (i.e., strong base-chain mixing gives larger polydisc). This completes the proof of Theorem 1.4. Remark 7.3 (Connection to weight-vector theorem). When the Markov chain is i.i.d. (all rows of P equal), the Markov cocycle reduces to the i.i.d. cocycle with $p = (P_{-1}, \ldots, P_{-N})$ (the common row). In this case, Theorem 1.4 recovers Theorem 1.1 (up to the equivalence between the $N \times N$ stochastic matrix P and the N-dimensional weight vector p). The Markov-chain setting thus strictly generalizes the i.i.d. setting, with the additional parameter $\rho_P$ controlling how far the chain is from i.i.d.

Lemma 10.1. (Lipschitz bounds for the Grassmannian action). For every $g \in GL(d,\mathbb{R})$ and every $V,W \in Gr(k,d)$ , <span id="page-29-0"></span> $$d_{\text{FS}}^{(k)}(gV, gW) \le ||g||^k ||g^{-1}||^k \cdot d_{\text{FS}}^{(k)}(V, W),$$ and for every $q, q' \in GL(d, \mathbb{R})$ and every $V \in Gr(k, d)$ , <span id="page-29-1"></span>(10.4) $$d_{FS}^{(k)}(gV, g'V) \le k \cdot \max(\|g^{-1}\|, \|g'^{-1}\|)^{k-1} \cdot \|g - g'\|.$$ Proof. For (10.3): by the singular-value characterization of $\|\Lambda^k g\|_{\text{op}}$ , we have $\|\Lambda^k g\|_{\text{op}} = \sigma_1(g)\sigma_2(g)\cdots\sigma_k(g) \leq \|g\|^k$ , where $\sigma_1(g)\geq\cdots\geq\sigma_d(g)$ are the singular values of g. Similarly $\|(\Lambda^k g)^{-1}\|_{\text{op}}\leq\|g^{-1}\|^k$ . The Lipschitz constant of the induced map on the unit sphere of $\Lambda^k\mathbb{R}^d$ (with the chord metric, which is bi-Lipschitz equivalent to $d_{\text{FS}}^{(k)}$ on Gr(k,d)) is bounded by $\|\Lambda^k g\|_{\text{op}}\cdot\|(\Lambda^k g)^{-1}\|_{\text{op}}$ , which is in turn bounded by $\|g\|^k\|g^{-1}\|^k$ . For (10.4): write $g_t = (1-t)g + tg'$ for $t \in [0,1]$ , and observe that $\Lambda^k g_t$ depends polynomially on t with $\partial_t(\Lambda^k g_t) = \sum_{j=1}^k \Lambda^{j-1} g_t \otimes (g'-g) \otimes \Lambda^{k-j} g_t$ (Leibniz rule on the exterior power). Taking the operator norm: $$\|\partial_t(\Lambda^k g_t)\|_{\text{op}} \le k \cdot \max(\|g_t\|)^{k-1} \cdot \|g - g'\|,$$ and therefore $$\|(\Lambda^k g)v_V - (\Lambda^k g')v_V\|_{\Lambda^k \mathbb{R}^d} \le k \cdot \max(\|g\|, \|g'\|)^{k-1} \cdot \|g - g'\|.$$ Projecting back to Gr(k,d) via the unit-sphere quotient, we obtain (10.4) with $\max(\|g^{-1}\|, \|g'^{-1}\|)$ in place of $\max(\|g\|, \|g'\|)$ after using the relation between the Fubini-Study metric on the Grassmannian and the chord metric on the unit sphere of $\Lambda^k \mathbb{R}^d$ , which contributes the factor $\max(\|g^{-1}\|, \|g'^{-1}\|)^{k-1}$ from the inverse direction. (This last step uses the fact that the unit-sphere projection map is bi-Lipschitz with constants involving $\|(\Lambda^k g)^{-1}\|_{\text{op}}^{k-1}$ , which is bounded by $\|g^{-1}\|^{k-1}$ as in the first part of the proof.) We adopt the abbreviation (10.5) $$\operatorname{ecc}^{(k)}(A) := \max_{1 \le i \le N} ||A_i||^k ||A_i^{-1}||^k = \operatorname{ecc}(A)^k.$$ 10.2. Strong k-irreducibility and the Markov operator on Gr(k,d). This subsection introduces the strong k-irreducibility hypothesis (Definition 10.2) and the induced Markov operator $P_{A,p}^{(k)}$ on $C^{\theta}(Gr(k,d))$ . The spectral gap of this operator on Hölder functions, recorded as Lemma 10.4, is the Grassmannian analog of Lemma 3.2 and is the input to the Kato perturbation argument of Subsection 10.5.3.

Lemma 10.4. (Spectral gap on Gr(k,d)). Let $A \in GL(d,\mathbb{R})^N$ be strongly k-irreducible, and let $p \in \Delta_N^{\circ}$ . Suppose the simplicity gap (10.1) holds. Then for every $\theta \in (0,1]$ there exist constants $C_^{(k)}(A,p,\theta) \geq 1$ and $\rho_^{(k)}(A,p,\theta) \in (0,1)$ such that the operator $P_{A,p}^{(k)}$ on $C^{\theta}(Gr(k,d))$ satisfies <span id="page-30-2"></span>(10.7) $$\|(P_{A,p}^{(k)})^n - \Pi_{A,p}^{(k)}\|_{\theta \to \theta} \le C_^{(k)}(A, p, \theta) \cdot \rho_^{(k)}(A, p, \theta)^n, \qquad n \ge 0,$$ where $\Pi_{A,p}^{(k)}\varphi = (\int \varphi \, d\eta_{A,p}^{(k)}) \, \mathbf{1}$ is the projection onto constants and $\eta_{A,p}^{(k)}$ is the unique $P_{A,p}^{(k)}$ -stationary probability measure on Gr(k,d). Explicit formulas are <span id="page-30-3"></span>(10.8) $$\rho_*^{(k)}(A, p, \theta) = e^{-\theta(\lambda_k(A, p) - \lambda_{k+1}(A, p))/2},$$ <span id="page-30-4"></span>(10.9) $$C_^{(k)}(A, p, \theta) = \frac{4\binom{d}{k} \operatorname{ecc}(A)^{2k}}{1 - \rho_^{(k)}(A, p, \theta)}.$$

Lemma 10.6. (Persistence of the spectral gap). Let $A^0 = (A_1^0, \ldots, A_N^0) \in GL(d, \mathbb{R})^N$ be strongly k-irreducible with the simplicity gap $\lambda_k(A^0, p^0) > \lambda_{k+1}(A^0, p^0)$ at $p^0 \in \Delta_N^\circ$ . Let $\theta \in (0, 1]$ . Then there exists an explicit radius <span id="page-31-0"></span> $$(10.11) r_{\text{persist}}^{(k)}(A^0, p^0, \theta) := \frac{1 - \rho_^{(k)}(A^0, p^0, \theta)}{8k \binom{d}{k} \operatorname{ecc}(A^0)^{2k-1} C_^{(k)}(A^0, p^0, \theta)} > 0$$ such that for every $A \in GL(d, \mathbb{R})^N$ with $||A_i - A_i^0|| < r_{\text{persist}}^{(k)}(A^0, p^0, \theta)$ for all i, and every p in a neighborhood of $p^0$ in $\Delta_N^{\circ}$ : - (a) A is strongly k-irreducible; - (b) the simplicity gap $\lambda_k(A, p) > \lambda_{k+1}(A, p)$ persists; - (c) the operator $P_{A,p}^{(k)}$ has a spectral gap on $C^{\theta}(Gr(k,d))$ with the same constants as (10.8)-(10.9), up to a factor of 2.

Theorem 10.8. (Quantitative analyticity of $\Lambda_k(A, p)$ ). Let $d \geq 2$ , $1 \leq k \leq d-1$ , $N \geq 1$ , and $\theta \in (0, 1]$ . Let $A^0 = (A_1^0, \dots, A_N^0) \in GL(d, \mathbb{R})^N$ be strongly k-irreducible with the simplicity gap $\lambda_k(A^0, p^0) > \lambda_{k+1}(A^0, p^0)$ at $p^0 \in \Delta_N^o$ . Then there exists an explicit polydisc radius $$(10.13) r_{\mathrm{H}}^{(k)}(A^{0}, p^{0}, \theta) := \min \left( r_{\mathrm{persist}}^{(k)}(A^{0}, p^{0}, \theta), \, r_{\mathrm{Kato}}^{(k)}(A^{0}, p^{0}, \theta) \right) > 0$$ such that the partial sum of the top-k Lyapunov exponents $$\Lambda_k(A, p) := \lambda_1(A, p) + \cdots + \lambda_k(A, p)$$ extends to a holomorphic function $$\widetilde{\Lambda}_k: D_{r_{\mathbf{u}}^{(k)}}(p^0) \times B_{r_{\mathbf{u}}^{(k)}}(A^0) \to \mathbb{C}$$ on the polydisc of radius $r_{\rm H}^{(k)}(A^0,p^0,\theta)$ in both the weight vector $p \in \mathbb{C}^N$ and the matrix coefficients $A \in {\rm GL}(d,\mathbb{C})^N$ , with $\widetilde{\Lambda}_k(p,A) = \Lambda_k(A,p)$ when restricted to the real domain. Here $r_{\rm persist}^{(k)}$ is given by (10.11) and $r_{\rm Kato}^{(k)}$ is the Kato resolvent radius, given in closed form by <span id="page-33-3"></span>(10.14) $$r_{\text{Kato}}^{(k)}(A^0, p^0, \theta) := \frac{1 - \rho_^{(k)}(A^0, p^0, \theta)}{8 C_^{(k)}(A^0, p^0, \theta) \cdot \binom{d}{k} \operatorname{ecc}(A^0)^k},$$ the analog of the Kato radius (9.4) (Section 3) on the Grassmannian. <span id="page-33-0"></span>Corollary 10.9 (Quantitative analyticity of individual sub-top exponents). Under the hypotheses of Theorem 10.8 applied at both level k and level k-1 (with $k \geq 2$ ), the individual sub-top Lyapunov exponent $$\lambda_k(A, p) = \Lambda_k(A, p) - \Lambda_{k-1}(A, p)$$ extends to a holomorphic function on the polydisc of radius $$r_{\text{individual}}^{(k)}(A^0, p^0, \theta) := \min(r_{\text{H}}^{(k)}, r_{\text{H}}^{(k-1)}) > 0.$$ Proof of Corollary 10.9. The difference of two holomorphic functions is holomorphic on the intersection of their domains of holomorphy. Theorem 10.8 applied at level k gives a holomorphic extension of $\Lambda_k$ on the polydisc of radius $r_{\rm H}^{(k)}$ , and Theorem 10.8 applied at level k-1 gives a holomorphic extension of $\Lambda_{k-1}$ on the polydisc of radius $r_{\rm H}^{(k-1)}$ . Their difference $\Lambda_k - \Lambda_{k-1} = \lambda_k$ is therefore holomorphic on the intersection, which is the polydisc of radius $\min(r_{\rm H}^{(k)}, r_{\rm H}^{(k-1)})$ . - <span id="page-33-2"></span>10.5. Proof of Theorem 10.8. The proof of Theorem 10.8 follows the same four-step structure as the proof of Theorem 1.1 in Section 4, with the projective Markov operator replaced throughout by the Markov operator $P_{A,p}^{(k)}$ on Gr(k,d), and the Furstenberg-Khasminskii formula replaced by its analog for the partial sum $\Lambda_k$ . - <span id="page-33-1"></span>10.5.1. Step 1: spectral gap of the complex Markov operator on Gr(k, d). The complex Markov operator $P_{A,p}^{(k)}$ on $C^{\theta}(Gr(k,d),\mathbb{C})$ is defined exactly as in (10.6) but with $A_i \in GL(d,\mathbb{C})$ and $p_i \in \mathbb{C}$ , and $V \in Gr(k,d,\mathbb{C})$ ranging over the complex Grassmannian (whose real points are Gr(k,d)). By Lemma 10.6, for (A,p) in the persistence neighborhood of $(A^0,p^0)$ , the operator $P_{A,p}^{(k)}$ has a spectral gap on $C^{\theta}(\operatorname{Gr}(k,d),\mathbb{C})$ with constants $\rho_^{(k)}$ and $2C_^{(k)}$ . The leading eigenvalue is simple, isolated, and depends holomorphically on (A,p) in the persistence neighborhood, with eigenprojection $\Pi_{A,p}^{(k)}$ also depending holomorphically. 10.5.2. Step 2: Furstenberg-Khasminskii formula for $\Lambda_k$ . For real (A, p) in the persistence neighborhood, the partial sum of the top-k Lyapunov exponents admits the Furstenberg-Khasminskii formula <span id="page-34-1"></span>(10.15) $$\Lambda_k(A, p) = \int_{Gr(k, d)} \int_{GL(d, \mathbb{R})} \log \frac{\|\Lambda^k g \cdot v_V\|_{\Lambda^k \mathbb{R}^d}}{\|v_V\|_{\Lambda^k \mathbb{R}^d}} d\mu(g) d\eta_{A, p}^{(k)}(V),$$ where $\mu = \sum_{i=1}^{N} p_i \delta_{A_i}, \ v_V \in \Lambda^k \mathbb{R}^d$ is a unit decomposable representative of V, and $\eta_{A,p}^{(k)}$ is the unique $P_{A,p}^{(k)}$ -stationary measure on Gr(k,d) (Lemma 10.4). For a derivation see (Viana, 2014, Theorem 6.9 and Corollary 6.10). Define the integrand function <span id="page-34-2"></span> $$\psi_{A,p}^{(k)}(V) := \sum_{i=1}^{N} p_i \log \frac{\|\Lambda^k A_i \cdot v_V\|_{\Lambda^k \mathbb{R}^d}}{\|v_V\|_{\Lambda^k \mathbb{R}^d}}.$$ Then $\psi_{A,p}^{(k)} \in C^{\theta}(Gr(k,d))$ with explicit Hölder constant $\binom{d}{k} \operatorname{ecc}(A)^k$ (proof analogous to Lemma 2.2 of Section 9). Equation (10.15) can therefore be rewritten as (10.16) $$\Lambda_k(A,p) = \int_{Gr(k,d)} \psi_{A,p}^{(k)}(V) \, d\eta_{A,p}^{(k)}(V).$$ <span id="page-34-0"></span>10.5.3. Step 3: holomorphic extension of $\eta_{A,p}^{(k)}$ . By the Kato perturbation theorem on the persistence neighborhood (using the spectral gap from Lemma 10.6 and the resolvent estimate (10.14)), the leading eigenprojection $\Pi_{A,p}^{(k)}$ depends holomorphically on (A,p) in the polydisc of radius $r_{\text{Kato}}^{(k)}$ . The dual eigenprojection $(\Pi_{A,p}^{(k)})$ then identifies the unique stationary measure $\eta_{A,p}^{(k)}$ with $(\Pi_{A,p}^{(k)})^\mathbf{1}$ acting on the constant function $\mathbf{1}$ , and we obtain a holomorphic extension $$\widetilde{\eta}_{A,p}^{(k)}:D_{r_{\mathrm{Kato}}^{(k)}}(p^0)\times B_{r_{\mathrm{Kato}}^{(k)}}(A^0)\to (\mathbb{C}^{\theta}(\mathrm{Gr}(k,d),\mathbb{C}))^*,$$ where the dual is taken with respect to the natural pairing between $C^{\theta}$ and its dual. The function $\psi_{A,p}^{(k)}$ depends holomorphically on (A,p) in the same polydisc (the entries of A enter through $\Lambda^k A_i$ , which is a polynomial of degree k in the entries; the entries of p enter linearly). By construction, $\widetilde{\psi}_{A,p}^{(k)} \in C^{\theta}(Gr(k,d),\mathbb{C})$ and the map $(A,p) \mapsto \widetilde{\psi}_{A,p}^{(k)}$ is holomorphic. 10.5.4. Step 4: holomorphic extension of $\Lambda_k(A, p)$ . Define the candidate holomorphic extension by $$\widetilde{\Lambda}_k(A,p) := \widetilde{\eta}_{A,p}^{(k)} \big(\widetilde{\psi}_{A,p}^{(k)}\big) \quad \text{for } (A,p) \in B_{r_{\mathrm{H}}^{(k)}}(A^0) \times D_{r_{\mathrm{H}}^{(k)}}(p^0),$$ i.e., the pairing of the dual eigenprojection with the integrand function. As a composition of holomorphic maps (with respect to the strong topology on the dual space), $\widetilde{\Lambda}_k$ is holomorphic on the polydisc of radius $r_{\rm H}^{(k)} = \min(r_{\rm persist}^{(k)}, r_{\rm Kato}^{(k)})$ . For real (A, p) in the persistence neighborhood, the formula (10.16) agrees with For real (A, p) in the persistence neighborhood, the formula (10.16) agrees with $\widetilde{\Lambda}_k(A, p)$ , by uniqueness of the Furstenberg-Khasminskii integral and the spectral gap. Therefore $\widetilde{\Lambda}_k$ extends $\Lambda_k$ holomorphically. This completes the proof. 10.6. Discussion. This subsection collects three remarks on the scope, limitations, and comparisons of Theorem 10.8: the analytic-vs-Hölder parallel with the companion paper Thiam (Nov. 2025), the open problem of removing strong k-irreducibility, and the comparison of the polydisc radius with the projective case of Theorem 1.6. Remark 10.10 (Scope of Theorem 10.8). Theorem 10.8 is the analytic counterpart of (Thiam, Nov. 2025, Theorem 11.9) (the Hölder sub-top theorem). Both rely on strong k-irreducibility, both rely on the simplicity gap at level k, and both prove a quantitative regularity statement (analyticity here, Hölder continuity there) for the partial sum $\Lambda_k$ . Recovering individual sub-top exponents $\lambda_k$ is by subtraction (Corollary 10.9), exactly as in the Hölder case. Remark 10.11 (What is still open at the sub-top level). The hypothesis of strong k-irreducibility is the standard sufficient condition for the existence of a spectral gap on Gr(k,d), but it is not optimal. The qualitative real-analyticity of sub-top Lyapunov exponents in GL(d) goes back to Peres' theorem Peres (1991) for finitely supported measures with simple Lyapunov spectrum (without an irreducibility hypothesis on individual levels). Whether the radius $r_{\rm H}^{(k)}$ can be improved by removing the strong k-irreducibility hypothesis, replacing it with a weaker condition (such as Zariski density of the support, or the existence of a Schottky pair at level k), is open. Quantitative bounds in the absence of strong k-irreducibility would require a substantially different argument, possibly via the avalanche principle applied to the cocycle on $\Lambda^k \mathbb{R}^d$ . Remark 10.12 (Comparison with the projective case). Comparing the constants of Theorem 10.8 with those of Theorem 1.6, we see that the polydisc radius $r_{\rm H}^{(k)}$ depends on ${\rm ecc}(A^0)^{2k}$ (versus ${\rm ecc}(A^0)^2$ in the projective case) and on the simplicity gap $\lambda_k - \lambda_{k+1}$ (versus $\lambda_1 - \lambda_2$ ). The radius therefore degenerates at the boundary of the simple-spectrum locus at level k (where $\lambda_k(A^0, p^0) - \lambda_{k+1}(A^0, p^0) \to 0$ ), in the same qualitative way as the top-exponent radius degenerates at $\lambda_1 = \lambda_2$ (Theorem 1.5 of Section 8).

Proposition 12.2. (Method-optimality of $r_*$ ). Within the Kato perturbation scheme of Definition 12.1, the polydisc radius $$r_ = \frac{1}{4NK_L_{\rm op}}$$ is optimal up to absolute constants. Specifically, any proof satisfying axioms (K1)-(K3) with inputs $(\tau, N_{\theta}, L_{\rm op})$ and producing a polydisc radius r' on which the perturbative Neumann series for the resolvent converges on $\Gamma_*$ satisfies $$r' \le \frac{C_{\mathrm{sch}}}{NK_*L_{\mathrm{op}}},$$ for some absolute constant $C_{\rm sch} > 0$ ; no improvement beyond the scaling $1/(NK_*L_{\rm op})$ is achievable without strengthening the inputs.

Proposition 12.4. (Spectral-gap obstruction to analytic continuation, conditional). Let $A \in GL(d, \mathbb{R})^N$ and $p^0 \in \Delta_N^{\circ}$ with simple top Lyapunov exponent. Define the complex spectral collapse set (12.2) $$\mathcal{Z}(A,p^0) = \Big\{ z \in \mathbb{C}^N : \sum_i z_i = 1, \text{ the operator } P_{A,z} \text{ on } C^\theta \\ \text{has multiple eigenvalues of maximal modulus} \Big\}.$$ Then: - (i) (Conditional, assuming $\mathcal{Z}(A, p^0) \neq \emptyset$ .) The analytic extension $\widetilde{\lambda}_+$ of Theorem 1.1 cannot, in general, be analytically continued across $\mathcal{Z}(A, p^0)$ ; if $z_* \in \mathcal{Z}$ is a point at which the maximal eigenvalue of $P_{A,z}$ becomes a multiple root of finite order, a branch-point singularity arises by (Kato, 1980, Chapter II, Section 1.7). - (ii) $\mathcal{Z}(A, p^0)$ does not intersect the polydisc $D_{r_/2}(p^0)$ , so $\operatorname{dist}(p^0, \mathcal{Z}(A, p^0)) \geq r_(A, p^0, \theta)/2$ whenever $\mathcal{Z}$ is non-empty. Remark 12.5 (Existence and structure of $\mathcal{Z}(A, p^0)$ ). The non-emptiness of $\mathcal{Z}(A, p^0)$ is a substantive question that we do not address here. Several observations: - (a) Since $P_{A,z}$ acts on the infinite-dimensional space $C^{\theta}(\mathbb{P}^{d-1})$ , the "characteristic polynomial" is not literally well-defined for an infinite-rank operator; we therefore phrase $\mathcal{Z}$ via the eigenvalue-isolation criterion of the Kato perturbation theory directly. - (b) In finite-dimensional approximations (e.g., truncating the Markov operator to its action on a finite-dimensional space of test functions), $\mathcal{Z}$ becomes the discriminant variety of the characteristic polynomial of the truncated operator and is non-empty by general algebraic-geometry arguments. The continuum limit of these finite-dimensional discriminants is conjecturally the full $\mathcal{Z}(A, p^0)$ , but a rigorous proof is open. - (c) In specific examples (e.g., N=2 matrices with explicit complexification), one can sometimes locate concrete singularities of $\widetilde{\lambda}_+$ and verify that they lie on $\mathcal{Z}$ . We therefore prefer to formulate Proposition 12.4 as a conditional statement: if $\mathcal{Z}$ is non-empty, then the polydisc radius $r_*$ provides a lower bound on the distance to the nearest singularity. The non-trivial direction of optimality (showing that $\mathcal{Z}$ is non-empty and close to $p^0$ ) is left open.

Proposition 12.9. (Bernstein-type sharpness of Cauchy bounds). Let $A \in GL(d, \mathbb{R})^N$ and $p^0 \in \Delta_N^\circ$ with simple top Lyapunov exponent. Let $r_{sh}(A, p^0)$ denote the sharp radius of analyticity of $p \mapsto \lambda_+(A, p)$ (the distance from $p^0$ to the nearest singularity of the analytic extension). For every $\varepsilon > 0$ , there exist multi-indices $\alpha$ of arbitrarily large $|\alpha|$ such that <span id="page-41-2"></span>(12.3) $$|\partial_p^{\alpha} \lambda_+(A, p^0)| \ge \alpha! \cdot \frac{M_{\rm sh}(A, p^0) \cdot (1 - \varepsilon)^{|\alpha|}}{r_{\rm sh}(A, p^0)^{|\alpha|}},$$ where $M_{\rm sh}(A,p^0)$ is a suitable bound on $\widetilde{\lambda}_+$ at the boundary of its maximal polydisc of analyticity. Hence the polynomial growth rate $\alpha!/r^{|\alpha|}$ in the Cauchy bound of Theorem 1.2 is of the same form as the sharp growth rate, with only the base radius r differing.

Definition 10.2. (Strong k-irreducibility). A tuple $A = (A_1, \ldots, A_N) \in GL(d, \mathbb{R})^N$ is strongly k-irreducible (with respect to a probability vector $p \in \Delta_N^{\circ}$ , equivalently with respect to the support of any $p \in \Delta_N^{\circ}$ ) if there is no finite union $V_1 \cup V_2 \cup \cdots \cup V_m$ of k-dimensional linear subspaces of $\mathbb{R}^d$ such that $$A_i(V_1 \cup \cdots \cup V_m) = V_1 \cup \cdots \cup V_m$$ for every $i = 1, \dots, N$ . For k=1, this recovers the classical strong irreducibility hypothesis of Furstenberg and Kifer (1983); Guivarc'h and Raugi (1985). For $k \geq 2$ , it is the natural extension to the Grassmannian setting and is the standard condition under which the induced random walk on Gr(k,d) has a unique stationary probability measure (Viana, 2014, Theorem 6.9).

Definition 10.3. (Markov operator on Gr(k,d)). For $A \in GL(d,\mathbb{R})^N$ and $p \in \Delta_N^{\circ}$ , define the Markov operator $P_{A,p}^{(k)}$ on C(Gr(k,d)) by <span id="page-30-5"></span> $$(10.6) \qquad (P_{A,p}^{(k)}\varphi)(V) := \sum_{i=1}^{N} p_i \,\varphi(A_i V), \qquad \varphi \in C(Gr(k,d)), \quad V \in Gr(k,d).$$ The space $C^{\theta}(Gr(k,d))$ of $\theta$ -Hölder functions on Gr(k,d) is defined exactly as in the projective case (Section 2), with $d_{FS}^{(k)}$ in place of the projective metric. The seminorm is $$[\varphi]_{\theta}^{(k)} := \sup_{V \neq W} \frac{|\varphi(V) - \varphi(W)|}{d_{\mathrm{FS}}^{(k)}(V, W)^{\theta}}, \qquad \|\varphi\|_{C^{\theta}}^{(k)} := \|\varphi\|_{\infty} + [\varphi]_{\theta}^{(k)}.$$