Asymptotics in infinite monoidal categories

Abel Lacabanne; Daniel Tubbenhauer; Pedro Vaz

arxiv: 2404.09513 · v2 · submitted 2024-04-15 · 🧮 math.CT · math.CO· math.RT

Asymptotics in infinite monoidal categories

Abel Lacabanne , Daniel Tubbenhauer , Pedro Vaz This is my paper

Pith reviewed 2026-05-24 02:38 UTC · model grok-4.3

classification 🧮 math.CT math.COmath.RT

keywords monoidal categoriesasymptoticstensor powersPerron-Frobenius theoryrandom walksindecomposable objectsGrothendieck ring

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The pith

Formulas describe the asymptotic growth of summands in tensor powers for monoidal categories with infinitely many indecomposables.

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

The paper supplies explicit formulas for the rate at which the number of summands grows under repeated tensor products in monoidal categories that may have infinitely many indecomposable objects. It applies generalized Perron-Frobenius theory together with random-walk methods on the set of indecomposables to obtain these rates. A reader would care because the formulas extend classical finite-category results to infinite settings and give concrete control over the complexity of tensor powers.

Core claim

Formulas exist for the asymptotic growth rate of the number of summands in tensor powers in certain finite or infinite monoidal categories, obtained via generalized Perron-Frobenius theory and random-walk techniques on indecomposable objects.

What carries the argument

Generalized Perron-Frobenius theory applied to the Grothendieck ring or fusion graph, paired with random walks on the indecomposable objects.

If this is right

The growth rate is determined by the largest eigenvalue of a suitable matrix or operator associated to the category.
Random-walk return probabilities on the graph of indecomposables yield the same asymptotic count.
The formulas recover known results when the category is finite.
The same techniques apply uniformly to both finite and infinite cases under the stated hypotheses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar growth formulas might hold in other algebraic structures equipped with a tensor product, such as certain Hopf algebras or fusion rings with infinite basis.
Numerical experiments on explicit infinite categories could test whether the predicted rates appear in low-dimensional examples.
The approach may connect to growth questions in representation theory of infinite groups or algebras.

Load-bearing premise

The monoidal categories under consideration admit the application of generalized Perron-Frobenius theory and can be analyzed via random walk techniques on their indecomposable objects.

What would settle it

A concrete infinite monoidal category in which the observed growth rate of summands in tensor powers deviates from the formula predicted by the generalized Perron-Frobenius and random-walk analysis.

read the original abstract

We discuss formulas for the asymptotic growth rate of the number of summands in tensor powers in certain (finite or infinite) monoidal categories. Our focus is on monoidal categories with infinitely many indecomposable objects, with our main tools being generalized Perron-Frobenius theory alongside techniques from random walks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends Perron-Frobenius and random walk methods to monoidal categories with infinitely many indecomposables, a straightforward next step with thin abstract details.

read the letter

The main point is that this paper discusses formulas for the asymptotic growth rate of the number of summands in tensor powers, now including monoidal categories that have infinitely many indecomposable objects. It relies on generalized Perron-Frobenius theory plus random walk techniques on those objects. That is the actual new piece: moving the existing finite-case methods into the infinite setting without claiming a full new framework. The tools line up with the problem, since random walks naturally track how tensor products decompose across indecomposables and Perron-Frobenius supplies the growth rates. The paper is honest in framing this as a discussion rather than a universal theorem, which keeps expectations in check. The approach does not look circular; it draws on established external results. The abstract gives no sample formulas or explicit conditions, which makes it hard to judge the precise scope or any convergence issues that might arise when the set of indecomposables is infinite. If the full paper supplies clear statements with verifiable hypotheses and derivations, that gap closes; otherwise the contribution stays mostly at the level of an extension rather than a sharp new result. This work sits squarely in monoidal category theory and its representation-theory applications. Specialists already familiar with the finite-case literature would get the most out of seeing how the methods carry over. It is grounded enough to merit peer review, though any referee would need to verify the infinite-case details carefully. I would send it out.

Referee Report

0 major / 1 minor

Summary. The manuscript discusses formulas for the asymptotic growth rate of the number of summands in tensor powers in certain (finite or infinite) monoidal categories. Its focus is on categories with infinitely many indecomposable objects, with the main tools being generalized Perron-Frobenius theory and random-walk techniques on those objects.

Significance. If the formulas are derived under clearly stated hypotheses, the work could usefully extend classical growth-rate results from finite monoidal categories to the infinite setting. The combination of Perron-Frobenius methods with random walks on indecomposables is a reasonable direction, though its payoff depends on concrete, verifiable applications.

minor comments (1)

[Abstract] The abstract states the focus and tools but supplies no derivations, conditions, or verification details. The introduction should supply at least one fully worked example (with explicit growth-rate formula) to make the claims assessable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report on our manuscript. The report contains no specific major comments to address point by point. We note the overall recommendation is uncertain and the significance section raises general points about hypotheses and applications, but without explicit queries we have nothing further to respond to at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external theory

full rationale

The paper's central discussion applies generalized Perron-Frobenius theory and random-walk techniques on indecomposable objects to derive asymptotic growth rates for summands in tensor powers. These tools are presented as external and independent (no load-bearing self-citations or fitted inputs renamed as predictions appear in the abstract or claim description). The approach is framed as exploratory under stated applicability hypotheses rather than a closed self-referential derivation, so the formulas do not reduce to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, invented entities, or additional axioms beyond the applicability of the named tools are stated.

axioms (2)

domain assumption Generalized Perron-Frobenius theory applies to the monoidal categories in question
Identified as the main tool in the abstract for deriving the growth rate formulas.
domain assumption Random walk techniques can be applied to analyze the tensor power decompositions
Listed alongside Perron-Frobenius as a core technique in the abstract.

pith-pipeline@v0.9.0 · 5566 in / 1248 out tokens · 21953 ms · 2026-05-24T02:38:05.760152+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel; phi_fixed_point echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

If X and the monoidal unit 1 are indecomposable, and we have X⊗X ≅ 1⊕X, then (b_n) is the Fibonacci sequence... dominating growth is ϕ^n for ϕ≈1.618 the golden ratio... action matrix ... has eigenvalues ϕ and −(ϕ−1)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

exponential growth theorem lim n→∞ n√b_n = PFdim_f X

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Growth Problems for Representations of Finite Monoids
math.RT 2025-02 unverdicted novelty 6.0

Conjecture expressing asymptotic growth of indecomposable summands in monoid-representation tensor powers via the Brauer character table of the group of units, with a proof under an extra hypothesis plus exact and asy...
Growth problems in diagram categories
math.RT 2025-03 unverdicted novelty 4.0

Derives asymptotic formulas for the growth rate of the number of summands in tensor powers of the generating object in semisimple diagram/interpolation categories.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · cited by 2 Pith papers · 9 internal anchors

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The Steinberg linkage class for a reductive algebraic group

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Andersen, P

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work page doi:10.1007/bf01245066
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On blocks of Deligne's category Rep(S_t)

URL:https: //arxiv.org/abs/0910.5695,doi:10.1016/j.aim.2010.08.010. [CEOT24] K. Coulembier, P. Etingof, V. Ostrik, and D. Tubbenhauer. Fractal behavior of tensor powers of the two dimensional space in prime characteristic. To appear inContemp. Math.URL:https://arxiv.org/abs/2405.16786. [COT24] K. Coulembier, V. Ostrik, and D. Tubbenhauer. Growth rates of ...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.aim.2010.08.010 2010
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On Tensor Products of Simple Modules for Simple Groups

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Soergel Calculus

URL:https://arxiv.org/abs/ 1309.0865,doi:10.1090/ert/481. [EGNO15] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik.Tensor categories, volume 205 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/ert/481
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Growth problems in diagram categories

URL:doi:10.1090/surv/205. [GT25] J. Gruber, and D. Tubbenhauer. Growth problems in diagram categories. To appear inBull. Lond. Math. Soc.2025. URL:https://arxiv.org/abs/2503.00685,doi:10.1112/blms.70163. [HS22] N. Harman and A. Snowden. Oligomorphic groups and tensor categories

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URL:https://arxiv.org/abs/2204. 04526. [HSS24] N. Harman, A. Snowden, and N. Snyder. The Delannoy category.Duke Math. J.173 (2024), no. 16, 3219–3291. URL: https://arxiv.org/abs/2211.15392,doi:10.1215/00127094-2024-0012. [KOK22] M. Khovanov, V. Ostrik, and Y. Kononov. Two-dimensional topological theories, rational functions and their tensor envelopes.Sele...

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URL:https://arxiv.org/abs/2011.14758,doi:10. 1007/s00029-022-00785-z. [KM16] T. Kildetoft and V. Mazorchuk. Special modules over positively based algebras.Doc. Math., 21:1171–1192,

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Special modules over positively based algebras

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URL:doi:10.1007/978-3-642-58822-8

One-sided, two-sided and countable state Markov shifts. URL:doi:10.1007/978-3-642-58822-8. [LTV23] A. Lacabanne, D. Tubbenhauer, and P. Vaz. Asymptotics in finite monoidal categories.Proc. Amer. Math. Soc. Ser. B, 10:398–412,

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[LTV24] A

URL:https://arxiv.org/abs/2307.03044,doi:10.1090/bproc/198. [LTV24] A. Lacabanne, D. Tubbenhauer, and P. Vaz. Code and more for the paper Asymptotics in infinite monoidal categories. 2024.https://github.com/dtubbenhauer/pfdim-inf-categories. [LSYZ22] C. Li, R.M. Shifler, M. Yang, and C. Zhang. On Frobenius–Perron dimension.Proc. Amer. Math. Soc., 150(12):...

work page doi:10.1090/bproc/198 2024
[12]

[Lus90] G

URL:https://arxiv.org/abs/1909.01693,doi:10.1090/proc/16034. [Lus90] G. Lusztig. Quantum groups at roots of1.Geom. Dedicata, 35(1-3):89–113, 1990.doi:10.1007/BF00147341. [Mis20] M. Mishna.Analytic combinatorics: a multidimensional approach. Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, [2020]©2020. [OEI23] OEIS Foundat...

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[13]

Tensor ideals in the category of tilting modules

URL:https: //arxiv.org/abs/q-alg/9611033,doi:10.1007/BF01234661. [Pól21] G. Pólya. Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz.Math. Ann., 84(1-2):149–160,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf01234661
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[Saw06] S.F

URL:doi:10.1007/BF01458701. [Saw06] S.F. Sawin. Quantum groups at roots of unity and modularity.J. Knot Theory Ramifications, 15(10):1245–1277,

work page doi:10.1007/bf01458701
[15]

Quantum Groups at Roots of Unity and Modularity

URL:https://arxiv.org/abs/math/0308281,doi:10.1142/S0218216506005160. [STWZ23] L. Sutton, D. Tubbenhauer, P. Wedrich, and J. Zhu. SL2 tilting modules in the mixed case.Selecta Math. (N.S.), 29(3):39,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0218216506005160
[16]

[Tub24] D

URL:https://arxiv.org/abs/2105.07724,doi:10.1007/s00029-023-00835-0. [Tub24] D. Tubbenhauer. Sandwich cellularity and a version of cell theory.Rocky Mountain J. Math., 54(6):1733–1773,

work page doi:10.1007/s00029-023-00835-0
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Sandwich cellularity and a version of cell theory

URL:https://arxiv.org/abs/2206.06678,doi:10.1216/rmj.2024.54.1733. [Twe71] R.L. Tweedie. Truncation procedures for non-negative matrices.J. Appl. Probability, 8:311–320,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1216/rmj.2024.54.1733 2024
[18]

[VJ67] D

URL:doi: 10.2307/3211901. [VJ67] D. Vere-Jones. Ergodic properties of nonnegative matrices. I.Pacific J. Math., 22:361–386,

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URL:http:// projecteuclid.org/euclid.pjm/1102992207. ASYMPTOTICS IN INFINITE MONOIDAL CATEGORIES 21 A.L.: Laboratoire de Mathématiques Blaise Pascal (UMR 6620), Université Clermont Auvergne, Campus Universitaire des Cézeaux, 3 place V asarely, 63178 Aubière Cedex, France, www.normalesup.org/∼lacabanne, ORCID 0000-0001-8691-3270 Email address:abel.lacabann...

work page arXiv 2006

[1] [1]

The Steinberg linkage class for a reductive algebraic group

URL: https://arxiv.org/abs/1706.00590,doi:10.4310/ARKIV.2018.v56.n2.a2. [APW91] H.H. Andersen, P. Polo, and K.X. Wen. Representations of quantum algebras.Invent. Math., 104(1):1–59,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.4310/arkiv.2018.v56.n2.a2 2018

[2] [2]

Andersen, P

URL: doi:10.1007/BF01245066. [Bia93] P. Biane. Estimation asymptotique des multiplicités dans les puissances tensorielles d’ung-module.C. R. Acad. Sci. Paris Sér. I Math., 316(8):849–852,

work page doi:10.1007/bf01245066

[3] [3]

On blocks of Deligne's category Rep(S_t)

URL:https: //arxiv.org/abs/0910.5695,doi:10.1016/j.aim.2010.08.010. [CEOT24] K. Coulembier, P. Etingof, V. Ostrik, and D. Tubbenhauer. Fractal behavior of tensor powers of the two dimensional space in prime characteristic. To appear inContemp. Math.URL:https://arxiv.org/abs/2405.16786. [COT24] K. Coulembier, V. Ostrik, and D. Tubbenhauer. Growth rates of ...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.aim.2010.08.010 2010

[4] [4]

On Tensor Products of Simple Modules for Simple Groups

URL:https://arxiv.org/abs/1102.3447,doi:10.1007/s10468-011-9311-5. [EW16] B. Elias and G. Williamson. Soergel calculus.Represent. Theory, 20:295–374,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10468-011-9311-5

[5] [5]

Soergel Calculus

URL:https://arxiv.org/abs/ 1309.0865,doi:10.1090/ert/481. [EGNO15] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik.Tensor categories, volume 205 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/ert/481

[6] [6]

Growth problems in diagram categories

URL:doi:10.1090/surv/205. [GT25] J. Gruber, and D. Tubbenhauer. Growth problems in diagram categories. To appear inBull. Lond. Math. Soc.2025. URL:https://arxiv.org/abs/2503.00685,doi:10.1112/blms.70163. [HS22] N. Harman and A. Snowden. Oligomorphic groups and tensor categories

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1090/surv/205 2025

[7] [7]

URL:https://arxiv.org/abs/2204. 04526. [HSS24] N. Harman, A. Snowden, and N. Snyder. The Delannoy category.Duke Math. J.173 (2024), no. 16, 3219–3291. URL: https://arxiv.org/abs/2211.15392,doi:10.1215/00127094-2024-0012. [KOK22] M. Khovanov, V. Ostrik, and Y. Kononov. Two-dimensional topological theories, rational functions and their tensor envelopes.Sele...

work page doi:10.1215/00127094-2024-0012 2024

[8] [8]

1007/s00029-022-00785-z

URL:https://arxiv.org/abs/2011.14758,doi:10. 1007/s00029-022-00785-z. [KM16] T. Kildetoft and V. Mazorchuk. Special modules over positively based algebras.Doc. Math., 21:1171–1192,

work page arXiv 2011

[9] [9]

Special modules over positively based algebras

URL: https://arxiv.org/abs/1601.06975. [Kit98] B.P. Kitchens.Symbolic dynamics. Universitext. Springer-Verlag, Berlin,

work page internal anchor Pith review Pith/arXiv arXiv

[10] [10]

URL:doi:10.1007/978-3-642-58822-8

One-sided, two-sided and countable state Markov shifts. URL:doi:10.1007/978-3-642-58822-8. [LTV23] A. Lacabanne, D. Tubbenhauer, and P. Vaz. Asymptotics in finite monoidal categories.Proc. Amer. Math. Soc. Ser. B, 10:398–412,

work page doi:10.1007/978-3-642-58822-8

[11] [11]

[LTV24] A

URL:https://arxiv.org/abs/2307.03044,doi:10.1090/bproc/198. [LTV24] A. Lacabanne, D. Tubbenhauer, and P. Vaz. Code and more for the paper Asymptotics in infinite monoidal categories. 2024.https://github.com/dtubbenhauer/pfdim-inf-categories. [LSYZ22] C. Li, R.M. Shifler, M. Yang, and C. Zhang. On Frobenius–Perron dimension.Proc. Amer. Math. Soc., 150(12):...

work page doi:10.1090/bproc/198 2024

[12] [12]

[Lus90] G

URL:https://arxiv.org/abs/1909.01693,doi:10.1090/proc/16034. [Lus90] G. Lusztig. Quantum groups at roots of1.Geom. Dedicata, 35(1-3):89–113, 1990.doi:10.1007/BF00147341. [Mis20] M. Mishna.Analytic combinatorics: a multidimensional approach. Discrete Mathematics and its Applications (Boca Raton). CRC Press, Boca Raton, FL, [2020]©2020. [OEI23] OEIS Foundat...

work page doi:10.1090/proc/16034 1909

[13] [13]

Tensor ideals in the category of tilting modules

URL:https: //arxiv.org/abs/q-alg/9611033,doi:10.1007/BF01234661. [Pól21] G. Pólya. Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz.Math. Ann., 84(1-2):149–160,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf01234661

[14] [14]

[Saw06] S.F

URL:doi:10.1007/BF01458701. [Saw06] S.F. Sawin. Quantum groups at roots of unity and modularity.J. Knot Theory Ramifications, 15(10):1245–1277,

work page doi:10.1007/bf01458701

[15] [15]

Quantum Groups at Roots of Unity and Modularity

URL:https://arxiv.org/abs/math/0308281,doi:10.1142/S0218216506005160. [STWZ23] L. Sutton, D. Tubbenhauer, P. Wedrich, and J. Zhu. SL2 tilting modules in the mixed case.Selecta Math. (N.S.), 29(3):39,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0218216506005160

[16] [16]

[Tub24] D

URL:https://arxiv.org/abs/2105.07724,doi:10.1007/s00029-023-00835-0. [Tub24] D. Tubbenhauer. Sandwich cellularity and a version of cell theory.Rocky Mountain J. Math., 54(6):1733–1773,

work page doi:10.1007/s00029-023-00835-0

[17] [17]

Sandwich cellularity and a version of cell theory

URL:https://arxiv.org/abs/2206.06678,doi:10.1216/rmj.2024.54.1733. [Twe71] R.L. Tweedie. Truncation procedures for non-negative matrices.J. Appl. Probability, 8:311–320,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1216/rmj.2024.54.1733 2024

[18] [18]

[VJ67] D

URL:doi: 10.2307/3211901. [VJ67] D. Vere-Jones. Ergodic properties of nonnegative matrices. I.Pacific J. Math., 22:361–386,

work page doi:10.2307/3211901

[19] [19]

URL:http:// projecteuclid.org/euclid.pjm/1102992207. ASYMPTOTICS IN INFINITE MONOIDAL CATEGORIES 21 A.L.: Laboratoire de Mathématiques Blaise Pascal (UMR 6620), Université Clermont Auvergne, Campus Universitaire des Cézeaux, 3 place V asarely, 63178 Aubière Cedex, France, www.normalesup.org/∼lacabanne, ORCID 0000-0001-8691-3270 Email address:abel.lacabann...

work page arXiv 2006