Scaling Supercritical Regime

• Scaling-supercritical regime is defined by system parameters exceeding critical levels, leading to explosive growth and emergent universal scaling laws.
• It spans applications from percolation to quantum fields, nonlinear PDEs and fluid mechanics, showcasing distinctive crossover behavior and anomalous statistics.
• Examining these regimes offers actionable insights, such as diagnosing critical exponents and designing experiments based on observable limit objects.

The scaling-supercritical regime refers, in diverse domains, to the set of phenomena, models, and mathematical behaviors arising when key parameters or nonlinearities cross a critical threshold, inducing new scaling laws, limiting objects, or anomalous statistical features. Systems studied in supercritical scaling regimes include stochastic processes (e.g., percolation, Hawkes processes, branching processes), nonlinear PDEs, quantum statistical systems, fluid mechanics, phase transitions, and condensed matter models. Across these fields, the supercritical regime is characterized by either explosive growth, universality governed by critical exponents, coalescent structures, anomalous fluctuations, or the emergence of new collective or limiting behaviors, often supporting scaling limits with distinctive mathematical properties.

1. Fundamental Definitions and Universal Features

In classic percolation theory and branching processes, the supercritical regime is defined by the control parameter $p$ (or, in branching, the reproduction number) exceeding a critical threshold $p_c$; macroscopic connectivity, infinite clusters, or persistent mass emerges. In autoregressive time series, supercriticality is marked by a characteristic root outside the unit circle, causing explosive variance and nonstandard limit theorems (Barczy et al., 2021). In dynamical systems and fluid mechanics, the adjective "supercritical" typically refers to a regime where dissipation is insufficient to counteract destabilizing nonlinearities or driving forces—e.g., the SQG equation with dissipation exponent $\alpha<1/2$, or dynamos where the Rossby number or Reynolds number exceeds a bifurcation threshold (Seshasayanan et al., 2018, Bulut et al., 2019).

In statistical physics and critical phenomena, the scaling-supercritical regime exists above the second-order (continuous) phase transition point. Here, although no sharp phase boundary remains, the system exhibits universal scaling, power laws, and critical crossovers between distinct macroscopic behaviors—liquid-like, gas-like, or critical quantum states (Wang et al., 12 Jun 2025, Li et al., 2023, Cui et al., 21 Sep 2025, Lv et al., 29 Oct 2024). In quantum field theory and high-field limits of quantum electrodynamics (QED), supercritical scaling refers to regimes with control parameters (e.g., nonlinearity $\chi$ or field strength) greatly exceeding their characteristic values, producing new limiting distributions, sharply peaked observables, or non-Poissonian emission statistics (Tamburini et al., 2019).

2. Scaling Laws, Crossover Structures, and Universality

A pervasive feature of the supercritical regime is the emergence of universal scaling laws, often embodied by critical exponents inherited from underlying symmetry or universality classes. In percolation models, oriented percolation on $\mathbb{Z}^2_{\mathrm{even}}$ admits an explicit scaling limit: after centering and diffusive rescaling (removing deterministic drift $\alpha$ and rescaling space by $\sigma\sqrt{n}$), the collection of rightmost infinite open paths converges in law to the Brownian web, a universal object capturing one-dimensional coalescence (Sarkar et al., 2011). The scaling map $$S_{\alpha,\sigma,\varepsilon}(x,n)=((x-\alpha n)/(\sigma\sqrt{\varepsilon}), n\varepsilon)$$ defines the canonical renormalization.

Statistical mechanics systems above their critical point admit two distinct crossover lines ("Widom lines" or $L^{\pm}$), symmetrically rooted in Ising-model $Z_2$ symmetry (Cui et al., 21 Sep 2025, Li et al., 2023). Near criticality, these crossovers follow power laws: $$\Delta H^{\pm} \sim |\Delta T|^{\beta+\gamma},\quad \Delta m^{\pm} \sim |\Delta T|^\beta$$ with $\beta,\gamma$ the order parameter and susceptibility exponents (mean field: $\beta=1/2,\gamma=1$; 3D Ising: $\beta\simeq 0.3265,\gamma\simeq 1.237$). The emergent $Z_2$ symmetry enforces two distinct branches, whose locations and scaling exponents are universal, confirmed across black hole thermodynamics, liquid-gas (LGPT), liquid-liquid (LLPT) transitions, and supercritical fluid experiments (Wang et al., 12 Jun 2025, Li et al., 2023). Notably, a single Widom line is insufficient to respect universality and symmetry.

Supercritical fluids exhibit scaling collapse of thermodynamic observables across different compounds, demonstrating "corresponding states" behavior. For example, the order parameter (gas-like fraction $\pi$ or density $\rho$) satisfies: $f_t(\tau) = A_t \tau^{-1/\nu_t}$ with $\nu_t$ universal, and multiple fluids' isotherms collapse onto a master curve via appropriate scaling variables (Ha et al., 2019). In supercritical water, scaling laws govern two-phase thickness $h$ as $$h \sim \sqrt{q x / (\dot m R \Delta h_{\mathrm{in}}) (\Delta\rho/\rho_{\mathrm{pc}})}$$; the critical balance between buoyancy and inertia is captured by a Richardson-like number $Ri \sim 1$ (Tripathi et al., 2020).

3. Limiting Objects and Scaling Limits

Supercritical scaling regimes often admit limiting processes distinct from those at or below criticality. In oriented percolation, the Brownian web arises universally as the scaling limit of the system after shearing and diffusive rescaling, capturing the geometry of coalescing infinite open paths (Sarkar et al., 2011). In Hawkes processes with nearly unstable supercritical kernels, two scaling limits emerge depending on the scaling of $a_T-1$ with total time $T$: a deterministic explosion (law of large numbers) for $T(a_T-1)\to\infty$, and a stochastic integrated CIR (Cox-Ingersoll-Ross) process under $T(a_T-1)\to\lambda\in(0,\infty)$ (Liu et al., 29 Jul 2024). In super-Brownian motion with supercritical branching, the additive martingale $W_t(\lambda_0)$ vanishes; however, the properly rescaled process $\sqrt{t}W_t(\lambda_0)$ converges in probability to a nondegenerate limit proportional to the derivative martingale, revealing Seneta-Heyde scaling (Hou et al., 2021).

Random walks in supercritical percolation clusters display normal, diffusive fluctuations for strong bias (trap exponent $\gamma>2$) but anomalous, subdiffusive (fractional power-law) fluctuations when $1<\gamma<2$; the global scale is $O(n^{1/\gamma})$, with only the order of magnitude presently established (Bowditch et al., 2021). In quantum Ising magnets, a quantum supercritical regime arises when a longitudinal field $h$ drives the system above its quantum critical point, with crossovers at $T \propto h^{z\nu/(\beta+\gamma)}$ and divergent magnetocaloric effect $\Gamma_h \propto T^{-(\beta+\gamma)/(z\nu)}$ (Lv et al., 29 Oct 2024).

4. Anomalous Fluctuations, Instabilities, and Nonstandard Statistics

The supercritical regime is frequently marked by anomalous behavior not seen at or below threshold. In biased random walks on supercritical percolation, for $1<\gamma<2$ fluctuations about the ballistic mean are of polynomial order $n^{1/\gamma}$, instead of classical $\sqrt{n}$ scaling; the temporal sequence exhibits oscillations and nontrivial stable limiting behavior is conjectured but not proved (Bowditch et al., 2021). Autoregressive processes of order 2 in the supercritical case ($|\lambda_1|>1$) require nontrivial random scaling $C_n$ to obtain a meaningful limit for the least squares estimator; the limit distribution is Gaussian but concentrated on a single ray in parameter space (Barczy et al., 2021).

For forced surface quasi-geostrophic equations with supercritical dissipation ($\alpha<1/2$), linear instabilities persist to the nonlinear level. In the log-supercritical case, global regularity can be proved for arbitrary large initial data and forcing by nonlinear maximum-principle techniques, but the fractional supercritical case lacks such results at present (Bulut et al., 2019).

In high-field quantum electrodynamics, the supercritical QED regime ($\chi\gg 1$) produces sharply peaked photon emission spectra close to the initial electron energy, with total emission rate per particle scaling as $\chi^{2/3}$ (Tamburini et al., 2019). Multiple recoil-correlated emissions suppress the peak when average photon number exceeds unity, leading to non-Poissonian statistics.

5. Experimental Confirmation and Physical Context

Supercritical regime scaling laws have been extensively validated across theoretical models, numerical simulations, and experiment. In supercritical fluids, inelastic x-ray and neutron scattering (IXS, SANS) on argon and CO$_2$ reveal crossover points consistent with Ising-class exponents in thermodynamic scaling (Li et al., 2023). Density-based and volumetric-fraction scaling collapses have been confirmed in both simulated Lennard-Jones fluids and real CO$_2$, H$_2$O (Ha et al., 2019). Epidemic models (e.g., COVID-19 spread) are described by supercritical scaling laws between reproductive number $R_t$ or $T_d$ and cumulative case counts, with critical slowing down and clear identification of transition points in phase space (Campi et al., 2020).

Scaling limits to the Brownian web have been established for coalescing random walks, drainage networks, voter-model interfaces, and percolation clusters (Sarkar et al., 2011). Quantum magnets (CoNb$_2$O$_6$), black holes (Reissner-Nordström AdS), and mean-field LLPT models of water exhibit supercritical crossovers with scaling laws matching those predicted from universality principles (Wang et al., 12 Jun 2025, Lv et al., 29 Oct 2024).

6. Mathematical and Physical Implications

The scaling-supercritical regime continually reaffirms the breadth of universality across mathematics and physics. Fundamental consequences include:

• The necessity of two crossover lines (not a single Widom line) to respect Ising $Z_2$ symmetry in the scaling regime (Cui et al., 21 Sep 2025, Li et al., 2023).
• Appearance of limiting objects distinguished from subcritical and critical regimes, such as the Brownian web, CIR or stable Levy processes, and rank-one concentration of estimators.
• Emergence of anomalous scaling exponents, non-Gaussian fluctuations, and singular behavior at the thresholds.
• The possibility to identify and exploit fundamental scaling variables as experimental diagnostics and design tools (e.g., two-phase thickness in supercritical flow (Tripathi et al., 2020), magnetocaloric ratio in quantum cooling (Lv et al., 29 Oct 2024)).
• Ongoing challenges in understanding full limit distributions, proving regularity or blowup for supercritical PDEs, and identifying thresholds in complex multiscale systems.

A plausible implication is that future work across fields, from quantum matter to pattern forming PDEs and stochastic social/biological processes, will locate further manifestations of scaling-supercritical behavior. Emphasis on multi-crossover structure, precise scaling variable identification, and analytic derivation of limit objects will continue to shape theory and experiment alike.