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Tropical Analysis of the Asymptotics of the Perron-Frobenius Eigenvector

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that the asymptotic Perron eigenvector of exp(kA) is governed by the tropical eigenspace of A, with the limit selecting a distinguished point—the far endpoint when the eigenspace is a segment—and depending only on that…

desk verdict The paper's central conjecture is false; the useful fragment is a counterexample to an existing algorithm. read the letter →

arxiv 1908.08234 v1 pith:V7QG3RY3 submitted 2019-08-22 math.RA

Balazs Kustar This is my paper · ORCID
classification math.RA MSC 15A8015A1815A48
keywords tropicalanalysisPerron-Frobeniuseigenvectortransfermatrixmax-plusalgebraasymptoticeigenspacecriticaleigenvectorsHadamardpower
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether the zero-temperature limit of the Perron eigenvector of the transfer matrix exp(kA) can be read off from tropical data of A. It establishes that the normalized limit P_infinity(A) always lies in the tropical max-plus eigenspace, but that it need not be a critical eigenvector; the earlier single-critical-eigenvector characterization does not generalize. The paper's central conjectures are: when the tropical eigenspace is a line segment of the form (0, v1+alpha, ..., vn+alpha), the limit is the far endpoint (0, v1+beta, ..., vn+beta), and more generally P_infinity depends only on the tropical eigenspace, not on the other entries of A. A complete proof of these conjectures would let physicists and applied mathematicians compute asymptotic eigenvectors directly from a few extremal entries of the matrix.

What carries the argument

The load-bearing object is the limit map P_infinity(A) = lim_{k->infty} (1/k) log(L(exp(kA))), a point in tropical projective space $TP^{{n-1}}$, together with the tropical max-plus eigenspace sigma_{max+}(A), the set of tropical eigenvectors of A. The argument connects the two through Hadamard powers: L(exp(kA)) = L($M^{{(k)}}$) with M = exp(A), and the eigenvectors of $M^{{(k)}}$ are the k-th Hadamard powers of tropical eigenvectors of M, so (1/k) log of the Perron eigenvector lies in sigma_{max+}(A). The conjectures identify which point of the eigenspace the limit selects: the far endpoint when the eigenspace is one-dimensional, and in general a function of the eigenspace alone.

What would settle it

Compute P_infinity for a 3 by 3 matrix with a two-dimensional tropical eigenspace and check, using high-precision arithmetic past k = 30, whether the normalized eigenvector sequence oscillates between two accumulation points; two distinct subsequential limits would falsify the claimed well-definedness. Alternatively, find two matrices A and B with identical sigma_{max+} but different P_infinity, which would directly falsify Conjecture 3.0.2.

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Extended reading notes

Core claim

The central discovery is that the asymptotic normalized Perron eigenvector P_infinity(A) = lim_{k->infty} (1/k) log(L(exp(kA))) is controlled by the tropical eigenspace sigma_{max+}(A), not by individual matrix entries. After proving P_infinity(A) lies in sigma_{max+}(A), the paper gives numerical evidence that it converges to a point in the interior or on the boundary of the eigenspace, and states two conjectures. Conjecture 3.0.1 says that if every tropical eigenvector has the form (0, v1+alpha, ..., vn+alpha) with 0 <= alpha <= beta, then P_infinity(A) = (0, v1+beta, ..., vn+beta), i.e. the limit is the endpoint of the eigenspace segment. Conjecture 3.0.2 says that equality of tropical eigenspaces, sigma_{max+}(A) = sigma_{max+}(B), implies equality of limits, P_infinity(A) = P_infinity(B). The paper also reports that the Schur-complement algorithm from cited perturbation theory often predicts a non-Perron eigenvector or no vector at all, and gives a concrete counterexample where a predicted vector lies in the eigenspace but is not the limit.

Load-bearing premise

The load-bearing premise is that the normalized Perron eigenvector sequence (1/k) log(L(exp(kA))) has a single limit as k tends to infinity for every matrix considered; the paper treats this as an immediate observation from plots of the first roughly thirty iterates, with no proof or error bound.

Editorial extensions

If this is right

  • For matrices whose tropical eigenspace is a segment, the zero-temperature Perron eigenvector limit is determined by the eigenspace's endpoint, so perturbing entries that do not change that eigenspace leaves the limit unchanged.
  • The invariance conjectured in 3.0.2 would make P_infinity computable from the small set of critical entries that determine the tropical eigenspace, rather than from all n^2 entries.
  • The counterexample to the cited Schur-complement algorithm shows that standard min-plus perturbation theory cannot always be used for this problem; a correct characterization must select among eigenvectors by a different rule.
  • If both conjectures hold, the asymptotic transfer-matrix eigenvector, and hence low-temperature behavior of the one-dimensional spin system, becomes a function of tropical geometry alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension of the paper's outlook is to compute P_infinity for matrices with identical tropical eigenspaces but different critical-graph cycle lengths; if the limits differ, Conjecture 3.0.2 would need refinement.
  • For higher-dimensional tropical eigenspaces, the selected limit point may be a weighted combination of critical eigenvectors, with weights possibly tied to cycle mean costs; the line-segment case would then be the special case where the weights collapse to an endpoint.
  • If the eigenspace-only dependence is correct, analogous statements should hold in the min-plus and max-times tropical algebras via exponentiation and logarithms, potentially yielding a tropical notion of eigenvector for tensors, as the paper suggests.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the asymptotic behavior of the Perron eigenvector of the matrix exponential A_k = exp(kA) as k tends to infinity, aiming to characterize the limit P_∞(A) = lim_k (1/k) log L(A_k) in terms of the max-plus (tropical) eigenspace of A. It proves elementary results: Proposition 2.3.3 identifies the limit of the normalized Perron eigenvalue with the max-plus eigenvalue of A, and Proposition 2.4.1 claims that the normalized Perron eigenvector lies in the max-plus eigenspace. The paper then presents numerical experiments for 3x3 matrices with several critical classes, states Conjecture 3.0.1 (a characterization of P_∞(A) when the tropical eigenspace is a one-dimensional segment), states Conjecture 3.0.2 (P_∞(A) depends only on the tropical eigenspace), and discusses the failure of the Akian et al. algorithm in this setting. The main advertised result is the conjectured complete characterization.

Significance. The question addressed is natural and potentially useful for transfer-matrix asymptotics and tropical spectral theory. The paper contains a few correct elementary observations, and the availability of the code on GitHub is a plus. If the conjectures were true, they would indeed provide a robust characterization of the Perron-eigenvector limit for a wider class than the single-critical-class case. However, the central conjecture is not merely unproved: it is contradicted by an explicit 3x3 matrix given below. Since the abstract claims a 'complete characterization for a larger class of matrices', and the paper's own main tool is conjectural and false as stated, the significance of the paper in its present form is very limited.

major comments (3)
  1. [Section 3, Conjecture 3.0.1] The conjecture is false. Take A = [[0, -M, -M], [-M, -1, 0], [-M, 0, -1]] with M > 1. The max-plus eigenvalue of A is 0, with critical graph consisting of the self-loop at node 1 and the 2-cycle 2<->3. The max-plus eigenspace in TP^2 is exactly {(0, t, t): -M <= t <= M}, i.e. all eigenvectors have the form (0, -M+alpha, -M+alpha) for 0 <= alpha <= 2M. Thus the hypothesis of Conjecture 3.0.1 holds with beta = 2M, and the conjecture predicts P_∞(A) = (0, M, M). However, for A_k = exp(kA) = [[1, e^{-kM}, e^{-kM}], [e^{-kM}, e^{-k}, 1], [e^{-kM}, 1, e^{-k}]], symmetry gives x_2 = x_3, and the Perron equations yield x_1/x_2 = 2 e^{-kM}/s with s = (e^{-k} + sqrt(e^{-2k} + 8e^{-2kM}))/2 ~ e^{-k}. Therefore (1/k) log(x_2/x_1) -> M-1, so P_∞(A) = (0, M-1, M-1). For M = 2, this is (0, 1, 1), strictly inside the segment [-2, 2] and not the endpoint (0, 2, 2) predicted by the conjecture. This directly invalidates the claimed complete characterization.
  2. [Sections 2.4 and 3] The existence of the limit P_∞(A) is asserted as an 'immediate observation from the plots' after the iteration is cut off around k = 30. No proof or error bound is supplied, and the heuristic derivation of Proposition 2.4.1 (that the normalized Perron vector lies in the eigenspace in the limit) rests on an informal 'as k→∞' step rather than a proved convergence statement. Since both conjectures are formulated in terms of this limit, a rigorous existence or convergence statement, or at least a conditional formulation, is needed before the conjectures can be meaningfully tested.
  3. [Section 3, Conjecture 3.0.2] The assertion that σ_max+(A) = σ_max+(B) implies P_∞(A) = P_∞(B) is supported only by two numerical examples (Figures 8 and 9). No argument is given, and it is not a formal consequence of Conjecture 3.0.1. As this is a second load-bearing claim in the advertised complete characterization, it needs either a proof or an explicit statement that it is only numerical evidence. The current text does not supply either.
minor comments (4)
  1. [Section 1] There is a typo: 'convengence' should be 'convergence'.
  2. [Section 4] The phrase 'Proofs of the conjectures in section 4' is incorrect: the conjectures appear in Section 3, not Section 4. Also, 'asymtotics' should be 'asymptotics'.
  3. [References] The reference to 'Synchronization and linearity' lists 'B. Francois' as the first author and the year 2001; the correct authors are F. Baccelli, G. Cohen, G.J. Olsder, and J.-P. Quadrat, and the book was published by Wiley in 1992.
  4. [Section 3, Akian counterexample] The displayed predicted vector from Theorem 6.1 of [Akian et al., 2006] is typeset ambiguously, mixing exponents and weights in a single column. It should be presented separately as a weight vector and an exponent vector so that the subsequent arrows to TP^2 are clear.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's conjectures are empirical generalizations rather than derivations, and its proved results use standard non-circular background results.

full rationale

The paper's central assertions, Conjectures 3.0.1 and 3.0.2, are explicitly introduced as conjectures supported by numerical experiments, not as conclusions derived from prior assumptions. The proved statements (Lemma 2.3.2, Proposition 2.3.3, Proposition 2.4.1) rely on standard Perron-Frobenius theory and standard tropical algebra facts, and none of them fits a parameter to data and then repackages the fit as a prediction. The limit P_infinity(A) is defined independently as lim_{k->infinity} (1/k) log(L(A_k)), and the conjectures only assert where that independently defined limit lies; the eigenspace condition is not built into the definition of P_infinity. The statement that the limit is well-defined is justified by an immediate observation from plots, which is an empirical limitation rather than a circular argument. There is also no load-bearing self-citation chain: the cited works are standard or external to this author, and the conjectures are not justified by citation at all. Even if Conjecture 3.0.1 is false, as a skeptical counterexample might show, falsity is not circularity. Thus the manuscript exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper rests on standard Perron-Frobenius theory, standard tropical spectral results from cited work, and an unproven numerical observation that the limit P_infinity(A) exists. No free parameters are fitted to data and no new ontological entities such as particles, forces, or dimensions are introduced. The main logical burden falls on the unproven limit existence and on the conjectures themselves.

assumptions (4)
  • standard math Perron-Frobenius theorem for positive matrices
    Used in Section 2.3 to guarantee the existence and uniqueness of the positive Perron eigenvector for each Hadamard power of the positive matrix.
  • domain assumption Known tropical eigenvalue result: as h->0, h log(rho(A_h)) converges to the max-plus eigenvalue of A
    Invoked in the introduction and Proposition 2.3.3, attributed to Gaubert and Plus. The paper builds on this result rather than reproving it.
  • domain assumption Existence of the limit P_infinity(A) = lim_{k->infinity} (1/k) log(L(exp(kA)))
    Assumed from numerical observations in Section 3. The paper provides no proof that the sequence converges without subsequences, which is a load-bearing premise for both conjectures.
  • standard math Monotonicity of positive power maps on products of matrix entries
    Used in Lemma 2.3.2 to show that cycles achieving the maximal mean cost are preserved under Hadamard powers, a structurally simple real-analytic fact.

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Pith. "Pith review of Tropical Analysis of the Asymptotics of the Perron-Frobenius Eigenvector." pith.science (2026). https://pith.science/paper/V7QG3RY3

@misc{pith2026190808234,
  author       = {Pith},
  title        = {Pith review of: Tropical Analysis of the Asymptotics of the Perron-Frobenius Eigenvector},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V7QG3RY3}},
  note         = {Machine review of arXiv:1908.08234}
}
abstract

Asymptotic properties of matrices are, in general, difficult to analyze with classical mathematical techniques. In very specific cases, there is a well-known connection between the asymptotic behavior of a matrix's leading eigenvector and the corresponding "tropical" matrix, arising out of the $max$ and $min$ operations innate in tropical analysis. In this paper we examine a more general class of matrices, and explore the extent to which we can generalize the results using tropical techniques. We find that while the original results do not easily generalize, we can still make some useful statements about the asymptotic behavior in the general case, and can give a complete characterization for a larger class of matrices than previously examined.

Figures

Figures reproduced from arXiv: 1908.08234 by the authors.

Figure 1
Figure 1. Visualization of the tropical span of v1 = h 0 −2 −1 i and v2 = h 0 1 0 i 2.3 The Perron Eigenvalue The Perron-Frobenius Theorem [Perron, 1907]. Let A ∈ R n×n be a matrix with strictly positive entries. Then the following statements hold: 1. There is a positive real number r, such that r is an eigenvalue of A and for any other eigenvalue λ of A, |λ| < r. 2. The eigenspace associated to r is one-dimensional. 3. There… view at source ↗
Figure 4
Figure 4. A = h 0 −1 −1 −4 0 −1 −1 −1 −4 i P∞(A) = h 0 −0.5 −1 i [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Loss of machine precision creates undesired artifacts after convergence. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figures from the paper (2 more)
Figure 6
Figure 6. Figure 6: A = h 0 −3 −2 1 0 −1 2 1 0 i P∞(A) = h 0 3 4 i [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 8
Figure 8. Figure 8: A = h 0 −4 −2 1 0 −3 −1 −1 0 i P∞(A) =  0 1/3 2/3  [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

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Reference graph

Works this paper leans on

10 extracted references · 8 canonical work pages

  1. [1]

    Akian, R

    M. Akian, R. Bapat, and S. Gaubert. Asymptotics of the perron eigenvalue and eigenvector. Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Volume 327, Issue 11, 1998. URL https://www.sciencedirect.com/science/article/pii/S0764444299801372#aep-bibliography-id6

    work page 1998

  2. [2]

    Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem

    M. Akian, R. Bapat, and S. Gaubert. Min-plus methods in eigenvalue perturbation theory and generalised lidskii-vishik-ljusternik theorem. arXiv Mathematics e-prints, 2006. URL https://arxiv.org/abs/math/0402090v3

    work page Pith review arXiv 2006

  3. [3]

    Francois, G

    B. Francois, G. Cohen, G. J. Olsder, and J.-P. Quadrat. Synchronization and linearity an algebra for discrete event systems. Wiley, 2001

    work page 2001

  4. [4]

    Perron-Frobenius theorem for nonnegative multilinear forms and extensions

    S. Friedland, S. Gaubert, and L. Han. Perron-frobenius theorem for nonnegative multilinear forms and extensions. Linear Algebra and its Applications, Volume 438, Issue 2, 2013. URL https://arxiv.org/abs/0905.1626

    work page Pith review arXiv 2013

  5. [5]

    Gaubert and M

    S. Gaubert and M. Plus. Methods and applications of (max,+) linear algebra. Annual Symposium on Theoretical Aspects of Computer Science, pages 261--282, 1997. URL https://link.springer.com/chapter/10.1007/BFb0023465

    work page doi:10.1007/bfb0023465 1997

  6. [6]

    Maclagan

    D. Maclagan. Introduction to tropical algebraic geometry. Contemporary Mathematics, Tropical Geometry and Integrable Systems, 2012. URL https://arxiv.org/abs/1207.1925

    arXiv 2012

  7. [7]

    O. Perron. Zur theorie der matrices. Mathematische Annalen, 64 0 (2): 0 248–263, 1907. doi:10.1007/bf01449896

    doi:10.1007/bf01449896 1907

  8. [8]

    Richter-Gebert, B

    J. Richter-Gebert, B. Sturmfels, and T. Theobald. First steps in tropical geometry. Contemporary Mathematics, 377, 07 2003. doi:10.1090/conm/377/06998

    work page doi:10.1090/conm/377/06998 2003

Show all 10 references
  1. [9]

    Rudiger and M

    R. Rudiger and M. Morvan. STACS 97: 14th Annual Symposium on Theoretical Aspects of Computer Science, Lubeck, Germany, February 27-March 1, 1997: proceedings. Springer, 1997

    1997

  2. [10]

    S. Santra. Physics - advanced statistical mechanics, 2013. URL https://nptel.ac.in/courses/115103028/

    2013

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