🔬 condensed matter

Emergence of continuously varying critical exponents in coupled map lattice as an effect of quenched disorder

This paper demonstrates that introducing quenched disorder in the form of asymmetric couplings within a coupled map lattice model causes the system's critical exponents for the transition to an absorbing phase to vary continuously, deviating from the standard directed percolation universality class observed in the disorder-free limit.

Original authors: Priyanka D. Bhoyar, Govindan Rangarajan, Prashant M. gade

Published 2026-07-20

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CC0 1.0

Original authors: Priyanka D. Bhoyar, Govindan Rangarajan, Prashant M. gade

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where tiny, chaotic events can organize themselves into grand, predictable patterns. This is the realm of non-equilibrium physics, a branch of science that studies systems that are constantly changing and never quite settle down, like a bustling city street or a spreading wildfire. In this chaotic dance, scientists look for "universality classes." Think of these as the DNA of chaos: no matter if you are studying a forest fire, a spreading virus, or a traffic jam, if they belong to the same "class," they will behave in exactly the same way. They share the same "critical exponents," which are just fancy numbers that describe how fast things spread or die out. For decades, scientists believed these numbers were fixed rules of the universe, like the speed of light. If you changed the details of your system, the big picture numbers shouldn't change. But what if those numbers could wiggle and shift? What if the rules of the game changed just by tweaking a single dial?

This is the question tackled by Priyanka D. Bhoyar, Govindan Rangarajan, and Prashant M. Gade in their recent work. They decided to play a game with a well-known model of chaos called a Coupled Map Lattice (CML). Picture a long line of dominoes, but instead of just falling over, each domino is a tiny, chaotic computer that talks to its neighbors. In the standard version of this game, the dominoes talk to both their left and right neighbors equally. This setup is famous for belonging to the "Directed Percolation" (DP) universality class—a very strict club where the rules of decay and growth are set in stone. The authors introduced a twist: they made the connections between the dominoes "asymmetric." Imagine that for some dominoes, the signal to the right is loud and clear, while the signal to the left is a whisper, and for others, it's the opposite. They called this "quenched disorder," which is just a scientific way of saying "frozen-in randomness" that doesn't change as the game plays out.

The results of their simulations were a surprise that broke the old rules. When the connections were perfectly balanced (or completely one-sided), the system behaved exactly as expected, following the strict DP rules. But as soon as they introduced a mix of loud-right and loud-left connections, the "critical exponents" started to change. It wasn't a jump to a new, fixed rule; it was a smooth, continuous slide. As they adjusted the fraction of "right-biased" connections (a parameter they called p), the numbers describing how fast the chaos died out changed continuously. For example, when p = 0.1, the decay exponent δ was 0.034, but when they increased p to 0.5, δ shifted to 0.158.

The authors found that this system didn't just break the rules; it invented a new kind of behavior where the "DNA" of the chaos is fluid. In their simulations, the order parameter (the fraction of "turbulent" or active sites) didn't just die out in a standard way; it followed a power-law decay that depended entirely on how much disorder they introduced. They observed that the time it took for the system to settle down grew incredibly slow, suggesting that the "defects" or active spots were getting stuck in a tug-of-war between moving left and moving right. The authors suggest this strange behavior might be linked to the "eigenvalue spectrum" of the system—imagine the mathematical fingerprint of the connections changing shape from a perfect ellipse to a flat line as the disorder changed. While the system still showed power-law decay (a sign of critical behavior), the specific numbers describing it were unique to the amount of disorder, violating the idea that universality classes are rigid.

In short, this paper shows that by introducing a specific kind of frozen randomness into a chaotic system, you can create a scenario where the fundamental laws of the transition are not fixed, but vary continuously. It's as if the dominoes, instead of falling at a set speed, could fall at any speed depending on how you arranged the whispers and shouts between them. The authors simulated this on a lattice of 200,000 sites, averaging over hundreds of different random setups to be sure. They found that for any non-zero amount of this specific disorder, the system belongs to a new, unknown class where the critical exponents are not constants, but variables that dance to the tune of the disorder. This challenges the long-held belief that universality classes are unchangeable, suggesting that in the messy, disordered real world, the rules of chaos might be far more flexible than we thought.

Technical Summary: Emergence of Continuously Varying Critical Exponents in Coupled Map Lattice as an Effect of Quenched Disorder

Problem Statement
The paper investigates the robustness of universality classes in non-equilibrium absorbing phase transitions, specifically within the Directed Percolation (DP) class. While universality classes are typically defined by fixed critical exponents determined by dimension and symmetry, the authors explore how quenched spatial disorder affects these classes. Previous studies have shown that disorder can lead to activated scaling (logarithmic decay) or new universality classes, but the specific scenario of continuously varying critical exponents at the critical point in a one-dimensional (1D) non-equilibrium system remains a subject of interest. The authors aim to determine if introducing asymmetric, quenched disorder into a well-known DP model can induce a regime where critical exponents change continuously with the disorder strength, thereby violating standard universality.

Methodology
The authors utilize a 1D Coupled Map Lattice (CML) model originally studied by Janaki and Sinha, which exhibits a transition from a laminar (absorbing) state to a turbulent (active) state.

Base Model: The system consists of $N$ sites ( $N = 2 \times 10^5$ ) with variables $x_i(t)$ . The evolution is governed by a circle map $f(x) = x + \omega - \frac{k}{2\pi}\sin(2\pi x)$ with fixed parameters $k=3.2$ and $\omega=0.29$ . The order parameter is the fraction of turbulent sites $m(t)$ (where $x_i > 0.5$ ).
Disorder Implementation: Spatial quenched disorder is introduced via asymmetric coupling. The coupling parameter $\epsilon$ $ϵ$ is split into two components, $\epsilon_1 = \epsilon/2 + 0.1$ $ϵ_{1} = ϵ /2 + 0.1$ and $\epsilon_2 = \epsilon/2 - 0.1$ $ϵ_{2} = ϵ /2 - 0.1$ . Sites are randomly assigned as "Type A" (probability $p$ $p$ ) or "Type B" (probability $1-p$ $1 - p$ ).
- Type A sites couple more strongly to the left neighbor ( $\epsilon_1$ ) and less to the right ( $\epsilon_2$ ).
- Type B sites couple more strongly to the right ( $\epsilon_1$ ) and less to the left ( $\epsilon_2$ ).
- The parameter $p$ is varied from 0 to 0.5. $p=0$ represents completely asymmetric coupling (all activity driven left), while $p=0.5$ represents a symmetric average despite local asymmetry.
Simulation: The system is simulated for various $p$ values. Critical exponents are determined via finite-size scaling, off-critical scaling, and analysis of the power-law decay of the order parameter $m(t) \sim t^{-\delta}$ at the critical coupling $\epsilon_c$ . The eigenvalue spectrum of the connectivity matrix is also analyzed to correlate spectral changes with critical behavior.

Key Results

Violation of Standard Universality: For $p=0$ , the system belongs to the standard DP universality class with critical exponents $\delta \approx 0.159$ , $\beta \approx 0.253$ , $\nu_{\parallel} \approx 1.732$ , and $z \approx 1.58$ . However, for any $p > 0$ , the system exhibits continuously varying critical exponents.
Continuous Variation of Exponents:
- The decay exponent $\delta$ at the critical point changes continuously with $p$ . It increases from $\delta = 0.034$ at $p=0.1$ to $\delta = 0.158$ at $p=0.5$ .
- The exponent $\beta$ (governing the steady-state density $m(\infty) \sim |\epsilon - \epsilon_c|^\beta$ ) also increases continuously with $p$ , ranging from 0.125 to 0.607.
- The correlation length exponent $\nu_{\parallel}$ and the dynamic exponent $z$ show significant variation. Notably, $z$ becomes very large ( $z \approx 3.90$ at $p=0.5$ and likely higher for $p < 0.5$ ), indicating subdiffusive behavior where disturbances move much slower than in standard DP.
Power-Law Decay at Criticality: Unlike some disordered systems that exhibit logarithmic decay (activated scaling) at the critical point, this model maintains a power-law decay $m(t) \sim t^{-\delta}$ for all $p$ , but with an exponent that depends on the disorder fraction.
Eigenvalue Spectrum Correlation: The authors link the changing critical behavior to the eigenvalue spectrum of the connectivity matrix.
- For $p=0$ , the spectrum forms an ellipse in the complex plane.
- For $0 < p < 0.5$ , the spectrum develops a "hole" and the eigenvalues spread around the boundary rather than lying strictly on a curve.
- For $p=0.5$ , the spectrum becomes real (lying on a line segment), despite the local asymmetry, suggesting the connections are "symmetric on average."
Non-Universal Behavior in the Absorbing Phase: A range of power-law decay exponents is observed in the absorbing phase (below $\epsilon_c$ ) for a range of coupling parameters $\epsilon$ . This range of non-universal behavior shrinks as $p$ decreases.

Significance and Claims
The paper claims to demonstrate a unique case of non-universal critical behavior in a nonequilibrium system where the critical exponents vary continuously with the strength of quenched disorder.

New Universality Class: The authors suggest that the system does not fit into any known universality class (including DP or the activated scaling class) because the exponents are not fixed but are functions of the disorder parameter $p$ .
Distinction from Weak Universality: The behavior differs from "weak universality" scenarios where scaling relations are preserved while exponents vary. Here, all critical exponents ( $\delta, \beta, \nu_{\parallel}, z$ ) vary continuously, and the system does not adhere to standard hyperscaling relations in the conventional sense, although the authors note that $\nu_{\parallel} \approx \beta/\delta$ holds approximately.
Mechanism: The continuous variation is attributed to the "tussle" or competition between activity driven in opposite directions by the asymmetric disorder. This competition leads to slow-moving defects and ultra-slow relaxation times (large $z$ ), preventing the system from settling into a standard fixed point.
Theoretical Implication: The work highlights that in the presence of specific types of quenched disorder, the concept of a fixed universality class may break down, leading to a line of fixed points or continuously varying exponents, a phenomenon previously more common in equilibrium statistical mechanics (e.g., the eight-vertex model) but now observed in a driven non-equilibrium CML.

The authors conclude that this model provides a clear example of how spatially quenched disorder can fundamentally alter the critical dynamics of a system, leading to a regime where critical exponents are not universal constants but depend on the specific details of the disorder distribution.

Technical Summary: Emergence of Continuously Varying Critical Exponents in Coupled Map Lattice as an Effect of Quenched Disorder

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