REVIEW
Universal Scaling of Conditional Mutual Information in Classical Stochastic Dynamics
Statistical Mechanics · Submitted 26 Oct 2026 · 56 references
Pith's verdict: A genuinely new scaling statement about CMI dynamics, with a transparent derivation, a working numerical tool, and an exact temporal-CMI prediction; the soft spots are acknowledged and non-blocking.
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Referee report
The paper
The paper claims
Conditional mutual information obeys a universal scaling form in systems with a growing Markov length.
The reading
What carries the argument
The central object is the scaling ansatz for the CMI, Eq. (1). Its derivation rests on the self-similar joint distribution of domain walls (Eq.·6) in the coarsening regime: pt(i1,…,ik;n) = ℓ(t)^{-k} gk(i1/ℓ(t),…,ik/ℓ(t); n/ℓ(t)). This implies an interval entropy Ht(n) = -n ρ(t) log ρ(t) + h(n ρ(t)), from which the CMI scaling follows by translation invariance. For numerical verification, the paper introduces a Hankel-based truncation scheme that compresses the matrix-product-state bond dimension by retaining the virtual directions contributing most to finite-configuration probabilities, rather than Schmidt weights.
Extended reading notes
Core claim
The paper establishes a universal scaling form for the conditional mutual information (CMI) in one-dimensional classical stochastic dynamics whenever the Markov length diverges with time. The scaling reads It(A:C|B) = (1/ℓ(t)^2) f(|B|/ℓ(t)) for fixed finite endpoint regions A and C, with ℓ(t) a growing length scale and f a model-dependent function that decays exponentially for large arguments. The inverse-square prefactor follows directly from translation invariance and the fixed sizes of the endpoints, as shown by a Taylor expansion of the interval entropy derived from the self-similar joint distribution of domain walls. The form is verified numerically in the inactive phase and at the critical point of parity-conserving directed percolation, and analytically for temporal CMI in the symmetric simple exclusion process via Gaussian fluctuating hydrodynamics, which reproduces the exact prefactor 1/32 without fitting parameters.
Load-bearing premise
The derivation of the CMI scaling relies on the assumption that the joint distribution of domain-wall positions is exactly self-similar (Eq.·6), which is motivated by coarse-grained coarsening theory but not derived from the microscopic stochastic dynamics of the PCDP model.
Editorial extensions
If this is right
- The scaling form provides a universal diagnostic for mixed-state phase transitions: a diverging Markov length imprints a characteristic 1/ℓ^2 decay of CMI on top of the exponential tail.
- The same form applies to temporal CMI, linking memory time to a growing Markov time; in the SSEP the analytical prefactor 1/32 is exactly reproduced.
- The inverse-square scaling dimension is robust across models with different growth laws for ℓ(t)—diffusive coarsening (t^{1/2}), critical dynamics (t^{1/z}), and inference-driven growth (t in SSEP).
- The Hankel truncation method may be broadly applied to classical-master-equation simulations using MPS, extending the accessible time scales for probability observables.
- The analytical derivation of temporal CMI resolves a previously reported discrepancy in the scaling exponent β, showing β=4 (z=2) rather than β≈3.2, and attributes the difference to finite-size effects.
Reading between the lines
- If the scaling form is universal, it should hold in higher-dimensional stochastic dynamics whenever a diverging length scale exists, though the scaling dimension may differ from 2.
- The self-similar assumption (Eq.·6) is not derived from microscopic dynamics; verifying it directly through higher-order correlation functions would test the theory's foundation.
- The success of the Hankel truncation scheme suggests that other singular-value-based compression methods for classical probability distributions should target observable probabilities rather than state-vector norms.
Original abstract
Conditional mutual information (CMI) characterizes dependence between two regions conditioned on an intervening buffer. Its characteristic decay length, the Markov length, has recently been identified as a diverging length scale at mixed-state phase transitions. Here, we investigate CMI dynamics in one-dimensional classical stochastic models and propose a universal scaling form for systems whose Markov length diverges with time. We derive this form in the inactive phase of parity-conserving directed percolation, where domain-wall coarsening generates a growing length scale and a self-similar probability distribution. We introduce a Hankel-based truncation scheme that enables matrix-product-state simulations at long times and over large spatial regions. Using this method, we verify the proposed scaling both in the inactive phase and at the nonequilibrium critical point, where the dynamical critical exponent governs Markov-length growth. More broadly, the same scaling form captures cases in which collective inference sets the diverging Markov length. Despite these distinct mechanisms, the CMI between fixed microscopic regions exhibits a common scaling dimension of two. Finally, we extend this framework to temporal CMI, establishing an analogous relation between the strength of conditional dependence in time domain and its characteristic memory time.
The compiled record
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.
The abstract, sentence by sentence
Conditional mutual information (CMI) characterizes dependence between two regions conditioned on an intervening buffer. Its characteristic decay length, the Markov length, has recently been identified as a diverging length scale at mixed-state phase transitions. Here, we investigate CMI dynamics in one-dimensional classical stochastic models and propose a universal scaling form for systems whose Markov length diverges with time. We derive this form in the inactive phase of parity-conserving directed percolation, where domain-wall coarsening generates a growing length scale and a self-similar probability distribution. We introduce a Hankel-based truncation scheme that enables matrix-product-state simulations at long times and over large spatial regions. Using this method, we verify the proposed scaling both in the inactive phase and at the nonequilibrium critical point, where the dynamical critical exponent governs Markov-length growth. More broadly, the same scaling form captures cases in which collective inference sets the diverging Markov length. Despite these distinct mechanisms, the CMI between fixed microscopic regions exhibits a common scaling dimension of two. Finally, we extend this framework to temporal CMI, establishing an analogous relation between the strength of conditional dependence in time domain and its characteristic memory time.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Conditional mutual information (CMI) characterizes dependence between two regions conditioned on an intervening buffer. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s1 | Its characteristic decay length, the Markov length, has recently been identified as a diverging length scale at mixed-state phase transitions. | background 1.00 | 0.07 | 0.04 | asserted 0.99 |
| s2 | Here, we investigate CMI dynamics in one-dimensional classical stochastic models and propose a universal scaling form for systems whose Markov length diverges with time. | contribution 1.00 | 0.85 | 0.18 | asserted 0.85 |
| s3 | We derive this form in the inactive phase of parity-conserving directed percolation, where domain-wall coarsening generates a growing length scale and a self-similar probability distribution. | contribution 0.70 | 0.92 | 0.09 | asserted 1.00 |
| s4 | We introduce a Hankel-based truncation scheme that enables matrix-product-state simulations at long times and over large spatial regions. | method 0.98 | 0.95 | 0.03 | asserted 1.00 |
| s5 | Using this method, we verify the proposed scaling both in the inactive phase and at the nonequilibrium critical point, where the dynamical critical exponent governs Markov-length growth. | result 0.92 | 0.95 | 0.20 | asserted 1.00 |
| s6 | More broadly, the same scaling form captures cases in which collective inference sets the diverging Markov length. | interpretation 0.64 | 0.70 | 0.14 | asserted 0.98 |
| s7 | Despite these distinct mechanisms, the CMI between fixed microscopic regions exhibits a common scaling dimension of two. | result 0.84 | 0.85 | 0.42 | asserted 1.00 |
| s8 | Finally, we extend this framework to temporal CMI, establishing an analogous relation between the strength of conditional dependence in time domain and its characteristic memory time. | contribution 0.88 | 0.94 | 0.11 | asserted 1.00 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:2610.36058 by the authors.
References
Discussion
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