Markov length can diverge in systems whose universal physics is spatially Markovian

Tarun Grover; Yu-Hsueh Chen

REVIEW

Markov length can diverge in systems whose universal physics is spatially Markovian

Yu-Hsueh Chen, Tarun Grover

Statistical Mechanics · Submitted 8 Oct 2026 · 62 references

Pith's verdict: An interesting paper that identifies cutoff-suppressed power-law CMI from irrelevant perturbations, but the central scaling argument rests on an unproven UV-finiteness assumption that the authors honestly flag.

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CitationTarun Grover, Yu-Hsueh Chen. "Markov length can diverge in systems whose universal physics is spatially Markovian." arXiv:2610.11235 (2026). https://pith.science/paper/IBEWWVJY

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Claim paper

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Editor's note

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The main result is genuinely new: an irrelevant detailed-balance-breaking perturbation on a critical local Gibbs state generates power-law CMI that vanishes in the continuum limit but leaves an infinite Markov length at any finite cutoff. This refines the classification of mixed-state phases, since it shows that states flowing to the same RG fixed point can have different CMI tails. The paper does several things well. The perturbative derivation leading to Eq. (3) is clean. The Gaussian solvable model with explicit numerical verification (Fig. 2) makes the scaling concrete. The one-dimensional result (CMI decays faster than l_B^{-2} iff the fixed point is Markovian) is a nice standalone constraint. The authors are transparent about their key assumption: CMI must be UV finite (footnote [14]). They also give a sufficient condition (rapid mixing) that would exclude such tails.

The soft spots are proportionate. The most load-bearing is the UV-finiteness assumption. For the perturbed non-Gaussian steady states that are the focus of the main claim, it is not proven. The examples cited in footnote [14] are ground states of QFTs or specific nonequilibrium systems, not the generic class considered here. The stress-test note is correct: if CMI were UV-divergent, the power-law tail could survive the continuum limit, undermatining the central claim. The authors acknowledge this, but they do not resolve it. The perturbative expansion also assumes that (e^{tL}Sigma)^x has non-zero overlap with scaling operators; this is plausible but unproven. The numerical check uses a single-site A and is confined to Gaussian models. The prediction for Toom's automaton is plausible but indirect.

These concerns do not negate the value of the paper. They are openly stated limitations of a first investigation. The paper is well written, the logic is clear, and the central distinction (continuum-surviving vs. cutoff-suppressed power-law CMI) is worth drawing.

Who is this for? Theorists working on mixed-state phases, CMI, RG, and nonequilibrium critical dynamics. It will be useful as a reference for the phenomenon.

Recommendation: send to peer review. The paper deserves a serious referee. The main soft spot should be discussed by the authors in revision; perhaps a comment on whether the UV-finiteness assumption can be checked in specific non-Gaussian models or relaxed. I would accept after minor revision.

Referee report

Major objections (2)

[§'Cutoff-suppressed power-law CMI from perturbing a critical Gibbs state', Eq. (4) and footnote [14]] The central scaling prediction (Eq. (4)) and the conclusion that the power-law CMI vanishes in the continuum limit rely on the assumption that CMI is UV-finite (lim_{a→0} CMI < ∞). The authors state this as an assumption and cite evidence from ground states of local QFTs and some nonequilibrium systems. However, the class of interest—critical Gibbs states perturbed by an irrelevant detailed-balance-breaking term—is not covered by these references. The Gaussian solvable model is linear and may not capture potential UV divergences from nonlinearities. The perturbative expression (Eq. (3)) involves a double time integral whose convergence in the continuum limit is not obvious, even with the × projection. The authors should either prove UV-finiteness under the stated dynamical assumptions (e.g., using properties of the generator L^{(0)} and the × projection) or delimit more precisely when it is expected to hold. Without this, the key claim that the power-law tail is cutoff-suppressed is not fully established for the generic non-Gaussian case.
[§'Cutoff-suppressed power-law CMI from perturbing a critical Gibbs state', paragraph beginning 'Being an irrelevant…] The argument that the operators (e^{tL^{(0)}}Σ)^× have nonzero overlap with scaling operators of the critical fixed point is described as an expectation ('One expects...'). While this is plausible, the claim of genericity (Abstract and §'Cutoff-suppressed power-law CMI from perturbing a critical Gibbs state') would be strengthened by a more concrete discussion of when the overlap is guaranteed or by an explicit example beyond the Gaussian model. As it stands, the genericity claim is not proven, though it is a reasonable physics expectation.

Minor objections (6)

[Abstract and Introduction] The phrase 'finite-range local classical Hamiltonian, whose CMI vanishes beyond the interaction range' is accurate for Gibbs states, but the paper might note that this Markov property holds specifically for the Gibbs distribution, not for arbitrary states derived from it.
[Eq. (3) and surrounding text] The notation ⟨·⟩_0 for the unperturbed average is used but could be defined explicitly in the main text (it appears in the derivation in Appendix A). Consider adding a brief definition after Eq. (3).
[Fig. 2 caption and inset] The main plot appears to be on a log-log scale, but this is not stated. The inset collapse is convincing; explicitly stating that the axes are logarithmic would improve readability.
[Appendix A, derivation of Eq. (A6)] The expansion of log(1+δ/Q) to second order and the justification that the linear term integrates to zero are correct, but a brief remark on the assumption that δ/Q is small in the relevant norm would be helpful for completeness.
[p.5, discussion of Toom's model] The estimate ω ≈ -1.3±0.2 for the Toom model is useful. The authors might briefly note the source of the uncertainty to guide future work.
[References] Reference [43] is listed as 'to appear'; if available, the authors should update the citation or note the arXiv number.

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The paper

The paper claims

Markov length can diverge in systems whose universal physics remains Markovian.

The reading

The paper shows that conditional mutual information (CMI) in critical Gibbs states of local Hamiltonians, when perturbed by an RG-irrelevant term that breaks detailed balance, acquires algebraic power-law tails. These tails cause the Markov length to become infinite at any fixed lattice cutoff, yet they are cutoff-suppressed: their amplitude vanishes as a positive power of the lattice spacing in the continuum limit, so the universal physics remains Markovian. This exposes a tension between standard RG universality—where states flowing to the same fixed point are considered equivalent—and mixed-state phase classification based on local reversibility, which treats any power-law CMI as an obstruction. The authors derive a general result for one-dimensional systems linking the CMI decay exponent to Markovianity, and they construct solvable Gaussian models and analyze the long-range Ising model to illustrate both cutoff-suppressed and genuine power-law CMI.

What carries the argument

The perturbative expansion of CMI around a Markov distribution, combined with the McLennan–Zubarev first-order expression for the steady state of Langevin dynamics with a detailed-balance-breaking perturbation. This yields Eq. (3): I(A:C|B) = (ε²/2) ∫₀^∞ dt ∫₀^∞ dt′ ⟨($e^{{tL^{(0)}}$}Σ)^× ($e^{{t′L^{(0)}}$}Σ)^×⟩₀, where $L^{{(0)}}$ is the backward generator of the equilibrium dynamics and Σ is the source generated by the perturbation. The irrelevant RG eigenvalue ω enters the scaling form CMI ∼ (ξ_cross/l_B)^{2|ω|} Φ(l_B/l). In one dimension, translational invariance gives I(A:C|B) ≃ (l/l_B)² g(l_B/a), with g(x)→0 if and only if the fixed point is Markovian.

Extended reading notes

Core claim

A generic irrelevant perturbation that breaks detailed balance on a critical local Gibbs state induces algebraically decaying conditional mutual information (CMI). The amplitude of this algebraic tail scales as a positive power of the lattice cutoff, so the CMI vanishes in the continuum scaling limit even though the Markov length is infinite at any finite cutoff. This cutoff-suppressed power-law CMI is distinct from genuine power-law CMI that survives the continuum limit and signals a genuinely non-Markovian fixed point. The distinction is essential for classifying mixed-state phases: cutoff-suppressed CMI obstructs local reversibility at any finite lattice spacing but disappears when the cutoff is removed, whereas genuine power-law CMI remains an obstruction even in the continuum.

Load-bearing premise

The scaling argument and the conclusion that the power-law tail vanishes in the continuum limit assume that the conditional mutual information is UV-finite (lim_{a→0} CMI < ∞); this is stated as an assumption and has not been proven for all considered systems.

Editorial extensions

If this is right

States flowing to the same RG fixed point can have qualitatively different Markov lengths, challenging the conventional RG notion of universality for mixed states.
Mixed-state phase classification based on local reversibility must distinguish cutoff-suppressed power-law CMI from continuum-surviving power-law CMI, since only the latter forbids a locally reversible path to a state with finite Markov length in the continuum.
The Ising-symmetric critical point of Toom's cellular automaton is predicted to exhibit cutoff-suppressed power-law CMI, providing a concrete test case for numerics.
In one dimension, the condition that CMI decays faster than 1/l_B² is equivalent to Markovian long-distance physics, giving a sharp diagnostic for spatial Markovianity.
The long-range Ising model offers a tunable example where the crossover from genuine to cutoff-suppressed CMI can be studied analytically and numerically.

Reading between the lines

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

The cutoff-suppressed CMI phenomenon may be generic in any critical system where an irrelevant operator breaks a symmetry (here, detailed balance). Similar behavior should occur in quantum systems under weak decoherence that breaks time-reversal symmetry.
The one-dimensional result provides a simple experimental or numerical test: if CMI in a translation-invariant state decays slower than l_B^{-2}, the fixed point is genuinely non-Markovian.
The paper suggests that mixed-state phase classification may need to incorporate 'dangerously irrelevant' parameters for CMI, analogous to dangerously irrelevant variables in equilibrium critical phenomena.
The distinction also implies that the Markov length alone is insufficient to characterize the phase; one must also determine whether the power-law tail survives the continuum limit.

Original abstract

Conditional mutual information (CMI) probes spatial non-Markovianity and plays a central role in a recently developed framework for defining mixed-state phases based on local reversibility. This framework sharply distinguishes exponentially decaying from algebraically decaying CMI, since the latter obstructs equivalence to a state with finite Markov length. Here we show that states that flow to the same renormalization-group (RG) fixed point can nevertheless exhibit qualitatively different CMI. Starting from a critical Gibbs state of a finite-range local classical Hamiltonian, whose CMI vanishes beyond the interaction range, we show that an RG-irrelevant perturbation that breaks detailed balance generically generates algebraically decaying CMI. Crucially, these power-law tails are cutoff-suppressed: although they lead to an infinite Markov length at every fixed lattice cutoff, they vanish as a positive power of the cutoff in the continuum limit. We construct solvable models that exhibit this phenomenon and predict that the Ising-symmetric critical point of Toom's cellular automata generically exhibits such behavior. We also find analogous cutoff-suppressed CMI in the high-temperature paramagnetic phase of the long-range Ising paramagnet, in contrast with the genuine power-law CMI at the finite-temperature critical point in this same model that survives the continuum limit. We also derive a general result that in one spatial dimension, CMI decays faster than the inverse square of the buffer length if and only if the long-distance physics is Markovian.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-09: 1 request, 6,556 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.48 · human participants 0.01 · controlled comparison 0.17 · randomized assignment 0.01 · states quantitative result 0.71 · compares with prior results 0.68 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Conditional mutual information (CMI) probes spatial non-Markovianity and plays a central role in a recently developed framework for defining mixed-state phases based on local reversibility. This framework sharply distinguishes exponentially decaying from algebraically decaying CMI, since the latter obstructs equivalence to a state with finite Markov length. Here we show that states that flow to the same renormalization-group (RG) fixed point can nevertheless exhibit qualitatively different CMI. Starting from a critical Gibbs state of a finite-range local classical Hamiltonian, whose CMI vanishes beyond the interaction range, we show that an RG-irrelevant perturbation that breaks detailed balance generically generates algebraically decaying CMI. Crucially, these power-law tails are cutoff-suppressed: although they lead to an infinite Markov length at every fixed lattice cutoff, they vanish as a positive power of the cutoff in the continuum limit. We construct solvable models that exhibit this phenomenon and predict that the Ising-symmetric critical point of Toom's cellular automata generically exhibits such behavior. We also find analogous cutoff-suppressed CMI in the high-temperature paramagnetic phase of the long-range Ising paramagnet, in contrast with the genuine power-law CMI at the finite-temperature critical point in this same model that survives the continuum limit. We also derive a general result that in one spatial dimension, CMI decays faster than the inverse square of the buffer length if and only if the long-distance physics is Markovian.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Conditional mutual information (CMI) probes spatial non-Markovianity and plays a central role in a recently developed framework for defining mixed-state phases based on local reversibility.	background 1.00	0.06	0.03	asserted 1.00
s1	This framework sharply distinguishes exponentially decaying from algebraically decaying CMI, since the latter obstructs equivalence to a state with finite Markov length.	background 1.00	0.11	0.07	asserted 1.00
s2	Here we show that states that flow to the same renormalization-group (RG) fixed point can nevertheless exhibit qualitatively different CMI.	result 0.78	0.88	0.48	asserted 1.00
s3	Starting from a critical Gibbs state of a finite-range local classical Hamiltonian, whose CMI vanishes beyond the interaction range, we show that an RG-irrelevant perturbation that breaks detailed balance generically generates algebraically decaying CMI.	result 0.88	0.94	0.26	asserted 1.00
s4	Crucially, these power-law tails are cutoff-suppressed: although they lead to an infinite Markov length at every fixed lattice cutoff, they vanish as a positive power of the cutoff in the continuum limit.	result 0.89	0.87	0.27	asserted 1.00
s5	We construct solvable models that exhibit this phenomenon and predict that the Ising-symmetric critical point of Toom's cellular automata generically exhibits such behavior.	contribution 0.56	0.91	0.15	hedged possibility 0.50
s6	We also find analogous cutoff-suppressed CMI in the high-temperature paramagnetic phase of the long-range Ising paramagnet, in contrast with the genuine power-law CMI at the finite-temperature critical point in this same model that survives the continuum limit.	result 0.99	0.95	0.08	asserted 1.00
s7	We also derive a general result that in one spatial dimension, CMI decays faster than the inverse square of the buffer length if and only if the long-distance physics is Markovian.	result 0.97	0.93	0.18	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
relations to other papers

levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Two geometries for the CMI

**Figure 2.** FIG. 2. CMI of the model in Eq. (

**Figure 3.** FIG. 3. CMI of the long-range Gaussian model in

Figures reproduced from arXiv:2610.11235 by the authors.

References

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Forward citations

Cited by 4 Pith papers

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

Long-range entangled criticality from coherent errors
quant-ph 2026-11 conditional novelty 8.0 of 10

Coherent errors in the honeycomb surface code induce a transition to a long-range entangled critical regime with divergent Markov length, coexisting with partial quantum memory.
Instability of Markov chains under infinitesimal coherent dissipation
quant-ph 2026-10 accept novelty 8.0 of 10

Strictly local Lindbladian dynamics can generate infinite Markov length from a product state, even when the dynamics is arbitrarily weak.
Thermodynamic cost of locally maintaining a steady state
cond-mat.stat-mech 2026-10 conditional novelty 7.0 of 10

Strictly local Gaussian dynamics can stabilize steady states with arbitrarily long-range conditional correlations, but any local implementation with finite-range equilibrium baths must dissipate entropy that grows wit...
Universal Scaling of Conditional Mutual Information in Classical Stochastic Dynamics
cond-mat.stat-mech 2026-10 accept novelty 7.0 of 10

Conditional mutual information in one-dimensional classical stochastic dynamics obeys a universal scaling form with an inverse-square prefactor when the Markov length diverges with time.

Discussion

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This is known to be true for many ground states of local quantum field theories [44–56], and has also been veri- fied for several nonequilibrium systems [4, 15, 39]. It is also consistent with the numerical observation in Ref. [57] that the CMI at the critical point of the dephased toric code is scale invariant. We point out that a nonlocal spacetime acti...

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