{"id":"967fe492-ec8f-4384-8656-875bdf378a93","arxiv_id":"1908.07469","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For i.i.d. products of invertible complex matrices, the normalized log of the spectral radius converges almost surely to the first Lyapunov exponent under a finite second moment, and in L^1 under a finite first moment, without any irreducibility assumption.","lead":"This paper proves that the largest eigenvalue of a product of independent random matrices grows at the same rate as the matrix norm: the strong law holds under a finite second moment, and a weak law under a finite first moment, with no irreducibility assumption. The result settles the spectral radius law of large numbers in its natural generality and supplies an example showing the i.i.d. assumption cannot be dropped.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; Proposition 2.5 is compressed but valid, and remaining issues are minor.","rationale":"I went through the proof skeleton of Theorems 1.1 and 1.2, focusing on the transferred large-deviation bound. The quotient argument in Proposition 2.5 is correct: for v∉Lμ, (2.4) gives the exact δ factor; since L_{π_*μ}={0}, every nonzero vector in V/Lμ has a.s. exponent λ1, so the cocycle average of π_*μ is unique and equals λ1, and Benoist-Quint's Theorem 2.4 applies with the stated uniform D_n. The later estimates in Lemmas 3.2-3.6 and Proposition 3.1 are derived from (2.2) in ways I checked; where exponents and indices need adjustment (e.g., blocks of length n-k rather than n), the changes are routine and do not affect the Borel-Cantelli or convergence-in-probability conclusions. The Markov-chain example of Section 4 also checks out: at times when the state is σ or ω, L_n is a conjugate of a generalized permutation matrix and has spectral radius 1, giving liminf 0, while visits to a give the limsup λ1. The remaining items (the 'Theorem 1.2' typo in the gap case, the sketched Proposition 3.8, and the implicit δ=1 convention when Lμ={0}) are minor and locatable; they do not change the verdict.","tokens_in":12600,"tokens_out":47149,"duration_ms":436052,"concrete_test":"Independently re-derive Proposition 2.5 from Theorem 2.4 for a concrete non-irreducible measure (e.g., block-diagonal μ with Lμ≠0), verifying (2.4) and λ1(π_*μ)=λ1(μ) line by line; if the derivation fails, the strong law in Section 3 does not follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern identified. The critical transferred large-deviation estimate, Proposition 2.5 (eq. (2.2)), is compressed but sound: the reduction to the quotient V/Lμ via (2.4) yields exactly the factor δ([v],[Lμ]), and the identities λ1(π_*μ)=λ1(μ) and L_{π_*μ}={0} follow because every nonzero vector in the quotient has a.s. Lyapunov exponent λ1 (otherwise it would lie in Lμ). Theorem 2.4 then gives the uniform lower bound with summable failure probability. The Section 3 estimates inherit this validity; the few places where parameter choices are implicit (e.g., applying Lemma 3.6 to a block of length m with ε' = ε n/m) are standard adjustments, not gaps. The only real defects are referee-level: in case (ii) of Theorem 1.1, 'Theorem 1.2' should read the already-proved case (i) applied to ⋀^s μ; Proposition 3.8 is presented as a sketch; and the convention δ(·,[Lμ])=1 when Lμ={0} is left implicit. None threatens the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves strong and weak laws of large numbers for the spectral radius of products of i.i.d. random matrices in GL_d(C). Theorem 1.1 shows that, under a finite second moment assumption, (1/n) log rho(L_n) converges almost surely to the first Lyapunov exponent lambda_1(mu); Theorem 1.2 shows that, under only a finite first moment, the same convergence holds in L^1 and hence in probability. No irreducibility assumption is imposed. The proof splits into three cases: a simple top Lyapunov exponent, a higher Lyapunov gap handled via exterior powers, and the case of equal exponents. The central estimates are transferred large-deviation bounds for the projective action (Proposition 2.5), together with geometric lemmas controlling the distance between the attracting point and the repelling hyperplane (Propositions 3.1 and 3.8). The paper also gives an ergodic stationary Markovian example in which the spectral-radius law of large numbers fails, showing that the i.i.d. assumption is essential.","tokens_in":12584,"tokens_out":19658,"duration_ms":197631,"significance":"If correct, the paper is a significant and clean contribution: it removes the strong-irreducibility assumption that was present in the earlier strong laws of Guivarc'h and Benoist--Quint, and it lowers the moment hypothesis to the optimal finite-first-moment range for the weak law. The proof is transparently structured and makes good use of known large-deviation results rather than introducing heavy new machinery. The counterexample in Section 4 is valuable because it demonstrates that the i.i.d. assumption, not just stationarity and ergodicity, is needed for the spectral-radius law. The paper also correctly credits the norm law of Furstenberg--Kesten and the large-deviation results of Benoist--Quint, and the reduction to the quotient space V/L_mu is a natural and effective device.","major_comments":[],"minor_comments":[{"comment":"In the sentence 'Applying Theorem 1.2 to eta', the reference should be to the already-proved case (i) applied to the measure eta, since Theorem 1.2 is the weak law and would not provide the almost sure convergence that the strong-law proof requires. This appears to be a typo, but it should be corrected.","section":"Section 4, proof of Theorem 1.1, case (ii)"},{"comment":"The proof of Proposition 3.8 is only sketched with 'one readily checks' and 'the proof of Proposition 3.1 applies verbatim.' Because this proposition is the key step leading to the weak law in Theorem 1.2, please expand at least the modifications in Lemmas 3.5 and 3.6, in particular the preservation of uniformity over H in H_{n,epsilon} when the failure probabilities D_n only tend to 0 rather than being summable.","section":"Section 3, Proposition 3.8"},{"comment":"The notation in (2.4) is ambiguous: the same symbol v denotes the original vector and its class in V/L_mu, and the displayed inequality is easy to misread. Please introduce separate notation for the class and the quotient norm, and state explicitly that the sequence D_n in (2.2) is uniform in v in V without 0; also record the convention that delta([v],[L_mu])=1 when L_mu={0}.","section":"Section 2.3, proof of Proposition 2.5"},{"comment":"The deduction of the estimates (3.10) and (3.11) from Lemma 3.3 is not immediate, because Lemma 3.3 is stated for the doubling comparison R_{2n} versus R_n while (3.10) compares lengths n and floor(n/2) and (3.11) compares lengths n and n-floor(n/2). Please add a sentence explaining the one-step or dyadic adjustment; the necessary estimate follows by the same arguments but should not be left implicit.","section":"Section 3, proof of Proposition 3.1"},{"comment":"The phrase 'a fortiori for every epsilon>0' is correct but deserves a short explanation: since the event delta(x^+_{L_n},H) <= e^{-epsilon n} shrinks as epsilon grows, the probability bound for small epsilon automatically gives the bound for all larger epsilon.","section":"Section 3, Lemma 3.6"}],"recommendation":"minor_revision","confidential_remarks":"I found no load-bearing mathematical error. The central estimates are sound, and the issues are expository: a misreferenced theorem in case (ii), a very compressed proof of Proposition 3.8, and some notational ambiguity in Proposition 2.5. The Markovian counterexample is a valuable addition. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know about 1908.07469. First, it settles a natural question: for i.i.d. products in GL_d(C), (1/n) log rho(L_n) -> lambda_1 almost surely under a finite second moment, and in L^1 under a finite first moment, with no irreducibility condition on the support. That genuinely goes beyond Guivarc'h 1990 and Benoist–Quint 2016, which both needed strong irreducibility and finite exponential moment. Second, the proof is not a black box; it builds on Benoist–Quint's large deviation estimates and splits into three cases by Lyapunov gap, and the main work is in the gap case.\n\nWhat the paper does well: the exterior-power reduction in case (ii) is elegant, and the Markov-chain counterexample in Section 4 is a nice touch, showing the i.i.d. assumption cannot be relaxed to stationary ergodic increments (you get liminf = 0 while limsup = lambda_1). The moment conditions match what the norm law would suggest, so the statements feel like the right ones.\n\nSoft spots, in proportion. Proposition 2.5 is the load-bearing piece and its proof is compressed: it extends [BQ16a, Prop 4.1] to the non-irreducible case in a paragraph. I read it carefully and the reduction to the quotient V/L_mu does work; the identities lambda_1(pi_*mu) = lambda_1(mu) and L_{pi_*mu} = {0} follow from Furstenberg–Kifer's results, and the stress-test note confirms the details. So this is not a gap, but in a final version I'd want a few more lines there. Second, in the proof of Theorem 1.1, case (ii), the text says 'Applying Theorem 1.2' where it should say the case already proved (Theorem 1.1, case (i)); that's a typo-level slip but worth fixing since the a.s. conclusion depends on it. Third, Proposition 3.8 is presented as a sketch ('we limit ourselves to indicate the proof'); it's clear how the lemmas adapt from summable error probabilities to probabilities tending to zero, but the details should appear in print.\n\nThe central argument holds up. This paper deserves a serious referee—I'd send it to an expert in random matrix products without hesitation. My own verdict would be accept after minor revision; the typos and compressed passages are fixable and don't threaten the main theorems. I'd cite this if I worked in the area.","headline":"A genuine and clean result: the spectral radius of i.i.d. matrix products satisfies the same law of large numbers as the norm, with no irreducibility assumption and essentially optimal moment conditions.","tokens_in":13397,"tokens_out":1719,"would_cite":true,"duration_ms":18361,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37H15","60F15","15A18","60B15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For i.i.d. products of complex matrices, the spectral radius has a law of large numbers: almost sure growth at the first Lyapunov exponent under finite second moment, convergence in probability under finite first moment, with no…","keywords":["law of large numbers","spectral radius","random matrix products","Lyapunov exponents","large deviations","no irreducibility assumption","projective space","Markovian increments"],"falsifier":"Run the probabilities in Proposition 2.5 numerically for a concrete non-irreducible i.i.d. measure on $\\mathrm{GL}_2(\\mathbb{C})$ with $L_\\mu\\neq\\{0\\}$ and a Lyapunov gap, scanning a fine grid of directions $[v]$; if some sequence $v_n\\to [L_\\mu]$ gives non-summable probabilities of violating the lower bound, the proposition is false and Theorem 1.1 does not follow. Simulating the Section 4 Markov chain with i.i.d. factors drawn from its stationary marginal, while checking that the spectral-radius law then holds, would further confirm that dependence, not the marginal distribution, is what creates the oscillation.","tokens_in":12124,"feed_emoji":"🎲","tokens_out":12313,"duration_ms":118161,"temperature":0.7,"pith_summary":"This paper proves that the spectral radius of an i.i.d. random walk on $\\mathrm{GL}_d(\\mathbb{C})$ satisfies a law of large numbers: if $\\mu$ has finite second moment, then $(1/n)\\log\\rho(L_n)$ converges almost surely to the first Lyapunov exponent $\\lambda_1(\\mu)$, and if $\\mu$ has only finite first moment, the same quantity converges in $L^1$ and hence in probability. The main advance is that no irreducibility assumption is put on the group generated by the support of $\\mu$, in contrast with earlier strong laws that required strong irreducibility and an exponential moment. The proof controls, with failure probabilities summable in $n$, the projective distance between the attracting point and the repelling hyperplane of $L_n$, then converts that distance into a lower bound for $\\rho(L_n)/\\|L_n\\|$ and combines it with the classical norm law. A section of counterexamples shows the i.i.d. assumption cannot be weakened to ergodic stationary Markovian increments: the same normalized spectral radius then has almost sure liminf $0$ and limsup $\\lambda_1(\\mu)$.","feed_headline":"Spectral radius of random products obeys a law of large numbers","feed_subtitle":"Finite second moment: almost-sure growth at the top Lyapunov exponent. Finite first moment: convergence in probability.","key_machinery":"The machinery is organized around the first Lyapunov gap---the integer $s$ with $\\lambda_1(\\mu)=\\cdots=\\lambda_s(\\mu)>\\lambda_{s+1}(\\mu)$---and around the subspace $L_\\mu$ of vectors whose growth rate is strictly below $\\lambda_1(\\mu)$. Projecting the walk to $V/L_\\mu$ makes the cocycle have a unique average, and Proposition 2.5 imports a large-deviation estimate: with summable failure probability, uniformly in $v\\notin L_\\mu$, $\\|L_n v\\|/\\|v\\|$ is at least $\\delta([v],[L_\\mu])e^{n(\\lambda_1(\\mu)-\\epsilon)}$ and at most $e^{n(\\lambda_1(\\mu)+\\epsilon)}$. The geometric heart is the pair $(x^+_{L_n},H^<_{L_n})$ in projective space---the fastest-expanding direction of $L_n$ and the hyperplane it repels---and the proof shows by a chain of large-deviation estimates (Lemmas 3.2--3.6) that this pair stays at distance at least $e^{-\\epsilon n}$ with high probability when the top exponent is simple. Lemma 2.2 then turns that distance into the ratio bound $\\rho(L_n)/\\|L_n\\|\\ge \\delta(x^+_{L_n},H^<_{L_n})/2$, reducing the problem to the known norm law; larger Lyapunov gaps are handled by applying the simple-gap result to the $s$-th exterior power, and the no-gap case by the inequality $a_d(g)\\le \\rho(g)\\le a_1(g)$.","core_discovery":"On the paper's own terms, the discovery is a pair of limit theorems with no irreducibility hypothesis: for i.i.d. factors on $\\mathrm{GL}_d(\\mathbb{C})$, $(1/n)\\log\\rho(L_n)\\to\\lambda_1(\\mu)$ almost surely when the measure has finite second moment, and in $L^1$ when it has finite first moment (Theorem 1.1 and Theorem 1.2). The same arguments give convergence of the full vector of eigenvalue moduli to the Lyapunov vector, work for right products as well as left products, and transfer to any local field, since the proofs use only exterior powers and projective geometry. The paper further claims that the i.i.d. condition is doing real work: for an ergodic stationary Markovian sequence of increments, the normalized spectral radius can oscillate, with liminf equal to $0$ and limsup equal to $\\lambda_1(\\mu)$, even though the norm still obeys the classical law.","pith_inferences":["One way to extend the paper's conclusion is to test whether the failure of the law for stationary Markovian increments persists when the stationary measure has full support on an irreducible but not strongly irreducible group; if it disappears, strong irreducibility rather than irreducibility is the relevant hypothesis for dependent increments.","A second extension suggested by the exterior-power reduction is to apply the same theorems to the spectral radii of all exterior powers $\\bigwedge^k L_n$, which would give a law of large numbers for the entire Lyapunov spectrum; the paper states only the eigenvalue-moduli version explicitly.","The strong-law question with finite first moment may be approachable by replacing the summable large-deviation sequence with a slower decay and using a softer almost-sure argument; nothing in the paper rules this out."],"forward_implications":["If Theorem 1.1 is right, the first-order growth of every eigenvalue modulus of an i.i.d. random matrix product is deterministic and equal to the corresponding Lyapunov exponent, without any irreducibility hypothesis.","Theorem 1.2 shows that the weak law needs no stronger moment hypothesis than the classical norm law, so the spectral-radius result does not require an extra moment price once convergence in probability is enough.","The exterior-power argument shows the same laws hold for all eigenvalue moduli simultaneously, which is Remark 1.3's vector-valued convergence; this also covers products with repeated top exponents.","Because Section 4's Markovian example has spectral-radius oscillations only for stationary dependent increments, any extension to non-i.i.d. walks needs extra hypotheses; the paper identifies this as the natural next problem.","As the paper notes, the almost-sure version under finite first moment remains open; the method's summable large-deviation estimates require the second moment."],"supporting_citations":[{"why":"Supplies the large-deviation estimates for linear cocycles; Proposition 2.5 extends its irreducible-case bound to vectors whose growth is measured from the slow subspace $L_\\mu$.","marker":"[BQ16a]"},{"why":"Lemma 14.14 is quoted as Lemma 2.2 and converts a lower bound on the projective distance between attractive point and repelling hyperplane into a lower bound for $\\rho(g)/\\|g\\|$.","marker":"[BQ16b]"},{"why":"Defines the slow subspace $L_\\mu$, gives the variational formula for $\\lambda_1(\\mu)$, and proves that the projected measure on $V/L_\\mu$ has the same top exponent with trivial slow subspace.","marker":"[FK83]"},{"why":"Independent source for the same $L_\\mu$ properties and the maximal-invariant-subspace characterization, used in the proof of Lemma 2.6.","marker":"[Hen84]"},{"why":"Provides the classical almost-sure law of large numbers for the norm, the baseline to which the spectral-radius ratio is compared in the final step.","marker":"[FK60]"},{"why":"Multiplicative ergodic theorem used in Lemma 2.6 to identify the random lower-expansion subspaces and transfer convergence from the filtration to the projective distance.","marker":"[Ose68]"},{"why":"Introduces the first-Lyapunov-gap approach that the paper adapts, and supplies the regularity of the stationary measure behind Lemma 3.6.","marker":"[AG19]"}],"fun_headline_variants":["Spectral radius LLN without irreducibility","Finite moments drive laws of large numbers for spectral radius","Strong law for spectral radius under second moment","i.i.d. matters: Markovian spectral radius oscillates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on Proposition 2.5, which says that once the distance from the starting direction to the slow subspace $L_\\mu$ is factored in, the probability of growing far below the top Lyapunov rate is summable in $n$ uniformly over all starting vectors; the paper presents this as a routine extension of a known irreducible-case estimate and does not verify the uniformity line by line, and the strong law's almost-sure conclusion fails if that uniformity breaks.","fun_headline_variants_meta":{"raw":{"variants":["Spectral radius LLN without irreducibility","Finite moments drive laws of large numbers for spectral radius","Strong law for spectral radius under second moment","i.i.d. matters: Markovian spectral radius oscillates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000735,"raw_usage":{"total_tokens":3198,"prompt_tokens":773,"completion_tokens":2425,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":389,"completion_tokens_details":{"reasoning_tokens":2362}},"tokens_in":389,"tokens_out":2425,"duration_ms":18799,"temperature":1.0,"reasoning_tokens":2362,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:23:25.387911+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the probabilities in Proposition 2.5 numerically for a concrete non-irreducible i.i.d. measure on $\\mathrm{GL}_2(\\mathbb{C})$ with $L_\\mu\\neq\\{0\\}$ and a Lyapunov gap, scanning a fine grid of directions $[v]$; if some sequence $v_n\\to [L_\\mu]$ gives non-summable probabilities of violating the lower bound, the proposition is false and Theorem 1.1 does not follow. Simulating the Section 4 Markov chain with i.i.d. factors drawn from its stationary marginal, while checking that the spectral-radius law then holds, would further confirm that dependence, not the marginal distribution, is what creates the oscillation.","supporting_citations":[],"review_version":1}