Gaussian fluctuations of non-reciprocal systems

Glasha Osipycheva; Peter B. Littlewood; Sergei Shmakov

REVIEW 3 major objections 5 minor 28 references

Gaussian fluctuations of non-reciprocal systems

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper shows that in the simplest linear stochastic model of two non-reciprocally coupled fields, the equal-time correlation functions diverge as $k^{-2}$ on ordinary critical lines, as $k^{-4}$ at critical exceptional points, and up…

desk verdict The exact Gaussian correlators and k^-6 divergence are new and useful, but the 'additional CEP' along the hyperbola for D1≠D2 is really a tangency effect, not an exceptional point. read the letter →

arxiv 2411.17944 v2 pith:4NTGU4Y5 submitted 2024-11-26 cond-mat.stat-mech

Sergei Shmakov , Glasha Osipycheva , Peter B. Littlewood This is my paper · ORCID

classification cond-mat.stat-mech PACS 05.40.-a64.60.Ht05.70.Ln

keywords non-reciprocalsystemsGaussianfluctuationscriticalexceptionalpointsequal-timecorrelationfunctionsfinite-momentuminstability1/fnoisereaction-diffusionlinearstochasticdynamics

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

The paper works out the exact Gaussian statistics of the simplest spatially extended, non-reciprocally coupled pair of fields: two linear stochastic reaction-diffusion equations with white noise and unequal couplings $j_{12} \neq j_{21}$. Its central finding is that the non-reciprocity parameter $\Delta = j_{12} j_{21}$ controls the entire stability portrait: it enlarges the stable region, creates exceptional lines where the two modes coalesce, and, where those lines meet a critical line, it sharpens the usual $k^{-2}$ divergence of the equal-time correlation function into $k^{-4}$ and, at isolated doubly tuned points, $k^{-6}$ divergences. The paper also shows that sufficiently strong non-reciprocity combined with unequal diffusion constants produces a finite-momentum instability, and that eliminating one field leaves the other with induced colored noise of 1/f type. Because the model is exactly solvable, it provides a linear reference point—analogous to the Gaussian model in equilibrium—for perturbative and renormalization-group studies of non-reciprocal systems.

What carries the argument

The load-bearing object is the two-field Gaussian action built through the Martin-Siggia-Rose–Janssen–De Dominicis procedure, from which the equal-time correlation functions (8) are computed exactly. The non-reciprocity parameter $\Delta = j_{12} j_{21} = j_+^2 - j_-^2$ controls the stability condition $m_1 m_2 > \Delta$ and organizes the phase diagram; because the coupling matrix is non-normal, exceptional points (where two eigenvalues and their eigenvectors coincide) appear whenever the argument of the square root in the dispersion (2) changes sign, and critical exceptional points are the intersections of those exceptional lines with critical lines. At such points the denominators of (8) gain extra powers of $k^2$, which is the entire mechanism behind the $k^{-4}$ and $k^{-6}$ enhancements. The same denominator analysis, applied to the momentum-shifted parameters $m_i + D_i k^2$, yields the finite-momentum instability, and the effective single-field action obtained by integrating out $\phi_2$ supplies the noise-kernel replacement that generates 1/f noise.

What would settle it

Numerically integrate the overdamped equations (1) on a finite lattice in $d = 1$ for $\Delta < 0$ with parameters placed on the critical line $m_1 = -m_2$, and measure the equal-time structure factor $S(k) = \langle |\phi_1(k,t)|^2 \rangle$; if $S(k)$ does not scale as $k^{-2}$ while the frequency-resolved spectrum shows a divergence at a nonzero frequency, the static-criterion claim fails. Independently, at the $\Delta = 0$ CEP $m_1 = m_2 = 0$, check the predicted $k^{-4}$ and $k^{-6}$ power laws of the exact expressions (8) by direct numerical solution; a mismatch in any exponent would refute the central enhancement claim.

Watch

Extended reading notes

Core claim

For the overdamped system (1), the exact equal-time correlation functions (8) show that along generic critical lines $m_1 m_2 = \Delta$ the denominator yields the standard $k^{-2}$ behavior of equilibrium Model A dynamics, so mild non-reciprocity is statistically indistinguishable from equilibrium criticality. At a critical exceptional point, where the eigenvalues and eigenvectors of the dynamical matrix coalesce at the same time as the system is critical, the denominator acquires an extra vanishing factor and the correlations diverge as $k^{-4}$; when an exceptional point sits at the intersection of two critical lines (for example $m_1 = m_2 = 0$ at $\Delta = 0$, or the corners in the $\Delta < 0$ phase diagram with equal diffusion constants), the divergence reaches $k^{-6}$. In the strongly non-reciprocal regime $\Delta < 0$, the stability region is expanded and an oscillatory region appears; along the critical line $m_1 + m_2 = 0$ the orbits are closed, and the correlation functions still show the same $k^{-2}$ divergence, though the paper states that the static diagnostic is questionable there. If the ratio of diffusion constants is large enough, the modes cross the critical line at finite momentum, giving a finite-momentum instability; at the two critical momenta the correlator diverges as $|k - k_c|^{-1}$, while tangency at a CEP gives $|k - k_c|^{-2}$. Integrating out one field replaces the noise kernel $B_1$ by $B_1 + j_{12}^2 B_2 / (\omega^2 + (m_2 + D_2 k^2)^2)$, which in dimensions $d \leq 3$ produces 1/f-type noise, and the inertial extension (14) preserves the same qualitative hierarchy of divergences.

Load-bearing premise

The classification of critical lines in the $\Delta < 0$ regime rests on using the equal-time correlation function, even on the oscillatory line $m_1 = -m_2$ where the paper itself says the static correlation function may be an inappropriate diagnostic; if static correlations mischaracterize that transition, the $k^{-2}$ divergence claimed there is not the relevant critical quantity.

Editorial extensions

If this is right

Perturbative and renormalization-group treatments that start from this linear theory must use the stronger $k^{-4}$ and $k^{-6}$ divergences at critical exceptional points to set their upper critical dimensions, not the equilibrium $k^{-2}$ value.
A system that is unstable under reciprocal couplings can be made stable by adding non-reciprocity, because the stability region in the $(m_1, m_2)$ plane widens as $\Delta$ decreases.
Strong non-reciprocity with sufficiently different diffusion constants produces a finite-momentum instability; the resulting patterns have a preferred length scale set by $k_c$, and near onset the correlator divergence is $|k - k_c|^{-1}$ (or $|k - k_c|^{-2}$ at a tangency with a CEP).
Any linear field coupled to a second diffusive field picks up an additional noise term whose spectrum is 1/f-like in $d \leq 3$; the 1/f tail extends to all frequencies as the system approaches a critical exceptional point.
The inertial generalization shows the same $\Delta$-controlled stability diagram, the same $k^{-2}/k^{-4}/k^{-6}$ hierarchy of correlations, and an additional CEP on the internal critical line when $\gamma_1 = \gamma_2$.

Reading between the lines

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

If the enhanced CEP divergences persist after nonlinearities are added, the non-reciprocal Allen-Cahn model and similar nonlinear extensions would have a lower upper critical dimension than equilibrium Model A, a difference that could be tested by comparing fluctuation corrections in $d = 3$.
The noise-kernel replacement suggests a generic experimental signature: in any two-species activator-inhibitor or excitatory-inhibitory system near criticality, the power spectrum of one species should show a 1/f component inherited from the other species even when the couplings are reciprocal.
The paper leaves the frequency-resolved correlation function on the oscillatory critical line $m_1 = -m_2$ uncomputed; a natural extension is to look for a spectral rather than static divergence there, which would connect the transition to time-crystalline order discussed for non-reciprocal Ising models.
The $k^{-4}$ and $k^{-6}$ scalings at CEPs could serve as a practical probe in neural and active-matter experiments: spatial correlations decaying faster than the equilibrium expectation would indicate that the system sits at a critical exceptional point.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper gives the exact Gaussian correlators for a minimal linear non-reciprocal two-field model with white noise and diffusion. That is genuinely useful: it provides a clean linear benchmark for MSR calculations and shows k^-4 and k^-6 divergences at double-tuned points, which I don't think are in the earlier literature. The stability diagram in terms of Delta is neat, and the finite-momentum instability mechanism is clearly explained. The effective noise calculation after integrating out one field is also a nice addition. There is no fitting and no circularity here; the calculation is straightforward but done carefully.

The main problem is the labeling of the 'additional CEP' along the hyperbola for D1≠D2. The condition D2 m1 + D1 m2 = 0 makes the second bracket in Eq. (9) scale as k^2, giving a k^-4 denominator. But this is not an exceptional point. On the critical line m1m2 = Delta, the eigenvalue-coalescence condition reduces to (m1+m2)^2 = 0, so true CEPs sit at m1 = -m2. For generic D1≠D2, the tangency point satisfies m1+m2 ≠ 0, so the evolution matrix is diagonalizable with one zero eigenvalue and one nonzero. The enhanced divergence is real, but it is a momentum-space tangency—the line of modes with slope D2/D1 grazing the critical hyperbola—not eigenvector coalescence. The paper's own definition of CEP in Sec. I agrees with this reading, so calling the tangency point a CEP is internally inconsistent. This matters because true CEPs have distinct RG physics, and conflating the two could mislead nonlinear extensions. It is a fixable terminology/interpretation issue, but it should be fixed.

Two smaller things. The paper uses the equal-time correlator on the oscillatory critical line m1+m2 = 0 in the Delta<0 regime, while acknowledging that static correlations might be inappropriate there. That needs a better justification. And the '1/f noise' claim is dimension-dependent: the integrated noise goes as omega^{(4-d)/2}, so it is exactly 1/f only in d=2. The paper says '1/f type' in the abstract, which is fair, but the wording in the body is a bit loose.

The central calculation holds up. I think this deserves a serious referee; a revision that corrects the CEP terminology and clarifies the static-correlator caveat would make it a solid reference.

Referee Report

3 major / 5 minor

Summary. The paper studies the simplest linear, spatially extended, stochastic two-field models with non-reciprocal couplings, in both overdamped and inertial forms. It derives exact equal-time correlation functions, maps stability regions as a function of the non-reciprocity parameter Δ, and identifies divergence laws of the correlations near critical lines: the usual k^-2 divergence, enhanced k^-4 and k^-6 divergences at certain points labeled critical exceptional points (CEPs), and a finite-momentum instability for strong non-reciprocity and unequal diffusion constants. It further claims that integrating out one field generates 1/f-type noise for the other. The manuscript is written as a reference-level classification of linear non-reciprocal Gaussian models, with explicit formulas that are internally consistent in their main algebraic steps.

Significance. If the claims are correct, the paper provides a useful exact solvable baseline for perturbative and RG treatments of non-reciprocal field theories, similar in spirit to the Gaussian fixed point in equilibrium critical dynamics. Its strengths are that the correlation functions are derived explicitly without fitting parameters, the stability conditions and denominator structures are transparent, and the paper makes falsifiable predictions about the momentum divergence exponents at special parameter values. However, three load-bearing points currently undercut the conclusions as stated: the frequency exponent in the 1/f noise calculation has the wrong sign, the 'additional CEP' on the critical hyperbola is not an exceptional point, and the equal-time correlation diagnostic is used on an oscillatory critical line despite the paper's own caveat that this diagnostic may be inappropriate there. These are correctable in revision but affect the paper's central classification claims.

major comments (3)

[Sec. II.B, Eq. (13)] The evaluation of the integrated-out noise term is not correct as written. Equation (13) states that ∫ d^d k j12^2 B2/(ω^2 + (m2 + D2 k^2)^2) ∼ j12^2 B2 D2^{-d/2} ω^{(4-d)/2} for d ≤ 3 and ω >> m2. Dimensional analysis of the integral gives instead a prefactor D2^{-d/2} ω^{(d-4)/2}: for d=1 this is ω^{-3/2}, for d=2 it is ω^{-1}, and for d=3 it is ω^{-1/2}. Thus the claimed generic '1/f noise' does not follow; only in d=2 is the spectrum exactly 1/f. The exponent sign error is load-bearing for the paper's final conclusion about colored noise, and the text and the inertial analogue in Sec. III should be corrected accordingly.
[Sec. II.B, around Eqs. (9) and (11)] The point on the critical hyperbola where D2 m1 + D1 m2 = 0 is called a 'CEP' (critical exceptional point), but it is not an exceptional point of the evolution matrix. The exceptional-point condition for Eq. (2) is (m1 − m2)^2 + 4Δ = 0; combined with the critical line m1 m2 = Δ this reduces to (m1 + m2)^2 = 0, so true CEPs have m1 = −m2. At the additional point with D2 m1 + D1 m2 = 0 and D1 ≠ D2, the eigenvalues are 0 and −(m1 + m2) with m1 + m2 ≠ 0, so the matrix is diagonalizable. The enhanced k^-4 divergence there is real but is a momentum-space tangency (the mode-stability line is tangent to the hyperbola), not eigenvector coalescence. This distinction matters because the paper uses CEP language to connect to known RG phenomenology; the text should rename or explicitly qualify these points and avoid claiming that all enhanced divergences are CEP effects.
[Sec. II.B, around Eq. (10)] The k^-2 divergence along the m1 + m2 = 0 line is computed from the static equal-time correlation function, even though the paper itself states in Sec. II.B that 'looking at the static correlation function might be inappropriate' for the oscillatory critical line in the Δ<0 regime. The manuscript does not reconcile this caveat with the subsequent claim that this line 'also exhibits the same divergence as the other critical lines.' If static correlations are not the correct diagnostic on this line, the claimed critical divergence there is not established. The authors should either justify the use of equal-time correlations in this regime or provide a frequency-dependent analysis, and should temper the classification of this line as a critical line with k^-2 behavior.

minor comments (5)

[Throughout] The frequency variable ω and the '1/f' notation are not defined consistently; if ω is angular frequency, the statement '1/f noise' should specify whether the power spectrum is ∝ 1/ω or 1/f, and the corrected exponents from Eq. (13) should be presented in a consistent notation.
[Section heading] The heading 'INER TIAL SYSTEMS' contains a typo and should read 'INERTIAL SYSTEMS'.
[Fig. 1 caption] The caption for Fig. 1(c) says that 'the critical lines all have k^-2 divergence in the correlators,' but the text immediately discusses points with k^-4 and k^-6 divergences on those lines; the caption should be qualified to distinguish generic points from the special points.
[Eq. (17)] The displayed denominator in Eq. (17) is typeset in a way that is hard to parse; adding explicit parentheses around the two brackets would improve readability.
[References] Reference [26] is listed as 'See Supplemental Material' but the paper does not indicate an ancillary file or a full citation; the reference should be completed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all predictions are derived from the linear stochastic model without fitted inputs.

full rationale

The paper's central results are obtained analytically from the model in Eq. (1) via the MSR action Eq. (6): the stability condition m1m2 > Delta, the equal-time correlation functions in Eq. (8), the denominator factorizations in Eqs. (9)-(11), the k^-2/k^-4/k^-6 divergence hierarchy, the finite-momentum instability, and the effective 1/f noise spectrum in Eqs. (12)-(13). No parameter is fitted to the predicted quantities; Delta is defined as j12j21 and used as a control parameter. Citations to the authors' earlier work [1,11,12,13] provide terminology ('non-reciprocal phase transitions,' 'critical exceptional points'), contextual statements, and interpretive remarks such as 'k^-4 divergence often associated with CEPs [11,12],' but none of these citations supplies a load-bearing mathematical step that replaces the paper's own derivation. The equal-time correlation caveat in Sec. II.B is an acknowledged limitation of the chosen diagnostic, not a circular step. Even the contested labeling of the hyperbola tangency point as a CEP would be a scientific-accuracy or nomenclature concern, not a reduction of a prediction to its input. The derivation chain is self-contained: given the linear stochastic equations and Gaussian white noise, the correlation functions and noise spectra follow by exact calculation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data; all model parameters are inputs. The calculation is self-contained except for standard field-theoretic machinery. The non-trivial model assumptions are the white-noise, no-cross-diffusion, Ito convention, the equal-time-diagnostic premise, and the gamma1=gamma2 restriction in the inertial case. The 1/f noise conclusion further assumes d<=3 and omega>>m2.

assumptions (6)

domain assumption Noise terms xi1 and xi2 are Gaussian white noise, are uncorrelated with each other, and there is no cross-diffusion.
Stated at Eq. (1): 'We assumed that the noises are uncorrelated and that there is no cross-diffusion for simplicity.' The exact correlator formulas (8) depend on these choices.
domain assumption Ito discretization is chosen for the stochastic dynamics.
Stated after Eq. (1): 'We pick the Ito discretization.' This fixes the MSR action used for the correlation functions.
standard math The MSR/De Dominicis-Janssen path integral is the correct framework for computing equal-time correlations of the linear system.
Used in Sec. II.B to derive the action (6) and the correlators (8); standard field-theoretic machinery.
domain assumption Equal-time correlation functions are the relevant criticality diagnostic.
Sec. II.B: 'it is therefore appropriate to look at the static, or equal time, correlation function'. The paper itself notes this 'might be inappropriate' on the oscillatory line, making this a load-bearing premise.
domain assumption For the inertial model, analytic results are restricted to gamma1=gamma2.
Sec. III.A: 'we will proceed with the gamma1 = gamma2 case analytically'; the unequal-damping case is asserted to work similarly but not shown.
domain assumption The 1/f noise argument assumes d <= 3 and omega >> m2.
Eq. (13) is derived under these conditions; the exact 1/f behavior holds only for d=2.

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Cite this review

Pith. "Pith review of Gaussian fluctuations of non-reciprocal systems." pith.science (2026). https://pith.science/paper/4NTGU4Y5

@misc{pith2026241117944,
  author       = {Pith},
  title        = {Pith review of: Gaussian fluctuations of non-reciprocal systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4NTGU4Y5}},
  note         = {Machine review of arXiv:2411.17944}
}

abstract

Non-reciprocal systems can be thought of as disobeying Newtons third law - an action does not cause an equal and opposite reaction. In recent years there has been a dramatic rise in interest towards such systems. On a fundamental level, they can be a basis of describing non-equilibrium and active states of matter, with applications ranging from physics to social sciences. However, often the first step to understanding complex nonlinear models is to linearize about the steady states. It is thus useful to develop a careful understanding of linear non-reciprocal systems, similar to our understanding of Gaussian systems in equilibrium statistical mechanics. In this work we explore simplest linear non-reciprocal models with noise and spatial extent. We describe their regions of stability and show how non-reciprocity can enhance the stability of a system. We demonstrate the appearance of exceptional and critical exceptional points with the respective enhancement of fluctuations for the latter. We show how strong non-reciprocity can lead to a finite-momentum instability. Finally, we comment how non-reciprocity can be a source of colored, $1/f$ type noise.

Figures

Figures reproduced from arXiv: 2411.17944 by the authors.

**Figure 2.** FIG. 2: Re/Im( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: Stability regions (light grey), regions with oscillations (dark grey) and CEPs (stars) for the inertial system, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: Typical phase plots for the overdamped system with ∆ [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5: Typical phase plots for the overdamped system with ∆ = 0. The dot corresponds to the fixed point. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6: Typical phase plots for the overdamped system with ∆ [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Fruchart, R

M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature 592, 363–369 (2021)

2021

[2] [2]

Y. Avni, M. Fruchart, D. Martin, D. Seara, and V. Vitelli, Dynamical phase transitions in the non- reciprocal ising model, (2024), arXiv:2409.07481 [cond- mat.stat-mech]

arXiv 2024

[3] [3]

Y. Avni, M. Fruchart, D. Martin, D. Seara, and V. Vitelli, The non-reciprocal ising model, (2024), arXiv:2311.05471 [cond-mat.stat-mech]

arXiv 2024

[4] [4]

Hanai, Nonreciprocal frustration: Time crystalline order-by-disorder phenomenon and a spin-glass-like state, Phys

R. Hanai, Nonreciprocal frustration: Time crystalline order-by-disorder phenomenon and a spin-glass-like state, Phys. Rev. X 14, 10.1103/physrevx.14.011029 (2024)

doi:10.1103/physrevx.14.011029 2024

[5] [5]

B. J. Cook, A. D. Peterson, W. Woldman, and J. R. Terry, Neural field models: A mathematical overview and unifying framework, Math. Neuro. and Appl. V olume 2, 10.46298/mna.7284 (2022)

doi:10.46298/mna.7284 2022

[6] [6]

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work page 1972

[7] [7]

Dayan and L

P. Dayan and L. F. Abbott, Theoretical neuroscience (The MIT Press, 2001)

work page 2001

[8] [8]

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R. Calvo, C. Martorell, G. B. Morales, S. D. Santo, and M. A. Mu˜ noz, Frequency-dependent covariance reveals critical spatio-temporal patterns of synchronized activ- ity in the human brain, (2024), arXiv:2403.15092 [q- bio.NC]

work page Pith review arXiv 2024