A theoretical study predicts that an idealized driven, nonreciprocal XY system in two dimensions can show long-range order under sufficiently long-ranged, scale-free quenched disorder, even though the corresponding clean finite-noise system is predicted to show only short-range order. The study asks whether quenched disorder, a spatial disorder pattern treated as fixed in the calculation, can counter the order-destabilizing effects of long-range noise. It also examines an active surface and three-dimensional model systems. The results are analytical predictions from idealized equations, not measurements.
The model's two kinds of randomness
The calculation combines a quenched disordered vector field with an annealed-noise term. In the XY version, the field represents a local phase; in the surface version, it represents local height. The equations add a stiffness or surface-tension term and a coupling between the field, the random vector field and spatial gradients. Two exponents describe how correlations vary across space: y for the quenched disorder and y-bar for the annealed noise. A separate parameter, mu, measures the relative strength of transverse and compressive disorder components. These are parameters of the model, not measured properties of a particular material or active system.
To handle the nonlinear problem, the authors use perturbative dynamic renormalization group at one-loop order, together with one-loop Feynman-graph corrections. In plain terms, the calculation tracks how the model's behavior changes across scales. The authors compare those results with a linear reference, which provides a baseline for the predicted order. The paper points to an appendix for the graphs used in the correction analysis.
The main split appears in two dimensions
The linear baseline is restrictive. In two dimensions, it predicts only short-range order for any positive noise-correlation exponent. In three dimensions, it predicts short-range order when that exponent is greater than 1, but long-range order when it lies between 0 and 1. The nonlinear two-dimensional XY calculation reports a different possibility: long-range order under sufficiently long-ranged quenched disorder, even in the finite-noise setting that is short-range ordered in the clean reference.
The active surface shows that ordering need not be a single yes-or-no property. In the two-dimensional phase diagram for mu below 1, the model includes positional long-range order with super-diffusive dynamics, orientational long-range order with positional short-range order and either super- or sub-diffusive dynamics, and a regime with short-range orientational order. The calculation therefore treats position and orientation as separate forms of order, rather than assuming they change together.
The predicted relaxation behavior also changes with the disorder balance. The dynamic exponent z, which describes how relaxation scales with distance, decreases with y when mu is greater than 1. When mu is below 1, it varies nonmonotonically and crosses the reference value 2 when y equals 1 minus mu. That makes mu a theoretical control parameter for the model's scaling and ordering regimes.
Three dimensions bring an unresolved boundary
In three dimensions, the renormalization-group analysis identifies a stable phase when y is greater than 1 and four times mu plus y minus 1 is positive. It identifies an unstable critical point when y is less than 1 and the same combination is negative. That point separates an asymptotically effectively noninteracting regime from a strong-coupling phase that the perturbative calculation cannot reach. The reported three-dimensional outcomes include either long-range or short-range order.
In three dimensions, the reported dynamic relation makes z depend on y and mu. The predicted z can be above or below 2, while the spatial scaling exponent chi can be negative or positive. In the paper's interpretation, those alternatives correspond to slower or faster diffusion and to long-range or short-range order. The scaling exponents of the strong-coupling phase remain unknown, leaving that part of the three-dimensional picture unresolved.
A result that depends on the model
The study also predicts continuously varying scaling rather than one fixed set of values. For fixed y and y-bar, the scaling exponents change continuously with mu in both two and three dimensions. It reports transitions between long-range and short-range order as mu changes while y and y-bar stay fixed. The result makes the balance between transverse and compressive disorder a theoretical tuning knob, not a universal constant.
Those findings come with a narrow evidence base. The work uses idealized hydrodynamic equations and one-loop perturbative analysis, so it offers model predictions rather than measured outcomes. The strong-coupling phase is outside the calculation's reach, and its scaling properties and exponents are not predicted. The results therefore describe the regimes allowed by the stated model, not what a real system must do.
The supplied document is an arXiv version 1 preprint dated 28 August 2026.
Paper data and sources
Original title: Correlated disorder versus correlated noise: Ordering in active systems
Authors: Sudip Mukherjee, Abhik Basu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
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