Temporal Decay: Inverse Square-Root Law

• Temporal decay is defined by a t⁻¹/² behavior arising from threshold singularities in spectral densities, providing a universal framework across various domains.
• The law is applied to quantum decay (affecting survival amplitudes), financial market impact (describing latent liquidity relaxation), and wave equations with singular potentials.
• Methodologies like Fourier analysis, Laplace transforms, and steepest-descent techniques yield precise asymptotic decay laws with significant experimental and practical implications.

The inverse square-root law for temporal decay describes a regime in which the response—be it a survival amplitude in quantum decay, a propagating signal in a dispersive medium, or a price impact in financial markets—relaxes with time as $t^{-1/2}$. This power-law decay stands in contrast to both exponential relaxation and faster power-law exponents, reflecting underlying threshold singularities or diffusive re-equilibration processes. The inverse square-root temporal decay emerges in diverse disciplines, each with precise mathematical formulation and domain-specific implications.

1. Mathematical Formulation and General Mechanism

The canonical model is the late-time decay of a quantity $A(t)$, which asymptotically takes the form: $A(t) \sim K\,e^{-i\omega_0 t}\,t^{-1/2}$ where $K$ is a complex amplitude set by the microscopic parameters and $\omega_0$ a characteristic frequency (or threshold energy). This law arises generically when the underlying spectral density $\rho(E)$ has a branch-point singularity at threshold: $\rho(E) \sim (E - E_{\rm th})^{-1/2}$, corresponding to a local exponent $\alpha = -1/2$ in the density of states or response kernel (Jiménez et al., 2018).

In the context of Fourier analysis, this result follows from integrating spectral components weighted by such a threshold singularity. Applying Watson's lemma or steepest-descent techniques to the Fourier or Laplace representation produces the universal $t^{-(\alpha+1)}$ power law, with $t^{-1/2}$ as the special case $\alpha=-1/2$ (Jiménez et al., 2018, Jiménez et al., 2024).

2. Quantum Decay and Survival Amplitudes

Threshold Singularities and Long-Time Behavior

For an unstable quantum state prepared at $t=0$, the survival amplitude $A(t)$ is given by a Fock-Krylov (Fourier) integral: $$A(t) = \int_{E_{\rm th}}^\infty \rho(E) e^{-i E t/\hbar} dE$$ where $\rho(E)$ is the energy density of states above threshold $E_{\rm th}$. For systems with $\rho(E) \sim (E-E_{\rm th})^{-1/2}$, typical for one-dimensional problems or van Hove singularities, the long-time tail is dominated by a branch point, resulting in

$A(t) \sim K\,e^{-iE_{\rm th}t/\hbar} t^{-1/2}$

(Jiménez et al., 2018). The survival probability at late times thus decays as $|A(t)|^2 \sim t^{-1}$, signaling a slow, non-exponential relaxation fundamentally rooted in the spectral structure at threshold.

Physical Realizations and Experimental Accessibility

While this inverse square-root law is strictly generic when the density of states or matrix element vanishes as $(E-E_{\rm th})^{-1/2}$, its physical manifestations in quantum decay are challenging to isolate experimentally. Transition to this regime is delayed far beyond observable timescales for narrow resonances (Jiménez et al., 2018, Jiménez et al., 2024). In particular, relativistic corrections and threshold effects modify the critical timescales and expose subtle quantum deviations from classical exponential decay and time-dilation predictions (Jiménez et al., 2024).

3. Inverse Square-Root Decay in Financial Market Impact

Empirical Law and Microstructural Propagators

In the context of financial markets, the inverse square-root decay law governs the temporal relaxation of price impact following market orders. Empirical analysis of the Tokyo Stock Exchange finds that, after a market order of size $q$ executed at time $t_j$, the cumulative impact measured at a later time $t_i>t_j$ is well described by

$$G(t_i - t_j) \simeq A\,(\Delta t)^{1/2}\,(t_i-t_j+s_0)^{-1/2}$$

where $\Delta t$ is the time between orders and $s_0$ a short-time regularization (Maitrier et al., 22 Feb 2025). This temporal propagator reflects the mere mechanical relaxation of latent liquidity: after an initial displacement, the order book profile diffuses back, with impact decaying as $(t-t_j)^{-1/2}$.

Universal Double Square-Root Structure

The observed macroscopic square-root law for metaorder impact, $I(Q) \propto \sqrt{Q}$ with $Q$ the total traded volume, is quantitatively explained as the convolution of:

• An immediate $\sqrt{q}$ impact per individual child order.
• An inverse square-root temporal decay of each child’s impact: $G(\tau) \propto \tau^{-1/2}$.

Summing over many independent child orders, each relaxing as $t^{-1/2}$, recovers (via convolution) the $\sqrt{Q}$ law for cumulative metaorder impact (Maitrier et al., 22 Feb 2025).

Mechanical versus Informational Models

The universal and robust $t^{-1/2}$ decay observed in both real and synthetic metaorders (i.e., with randomly shuffled trader IDs) strongly supports a purely mechanical origin rooted in latent order-book dynamics (locally-linear order book, or LLOB model), not in informational-Bayesian price formation models (Maitrier et al., 22 Feb 2025). This finding holds up to intermediate timescales (10–30 minutes), with only mild deviations for extremely large or small orders.

4. Inverse Square-Root Relaxation in Wave Equations with Singular Potentials

Price’s Law and Model Equations

For wave equations on Minkowski space with inverse-square potentials,

$\Box_g u + \frac{\digamma}{r^2} u = 0$

the pointwise decay of solutions exhibits two principal regimes at timelike infinity (Baskin et al., 2022):

• Zero-momentum regime (compact $r$): $u(t, r) \lesssim t^{-2-\beta(\digamma)}$.
• Nonzero-momentum regime (outgoing rays): $$u(t, \gamma t) \lesssim t^{-2-\frac{1}{2}\beta(\digamma)}$$.

Here, $\beta(\digamma) = \sqrt{1 + 4\digamma} - 1$ up to resonance corrections. For classical fields without potential ($\digamma=0$), the nonzero-momentum regime yields $t^{-3/2}$ decay. With more singular behaviors at the threshold (e.g., transmission resonances), the leading edge of the relaxation can shift exponents toward or beyond the $-1/2$ regime, depending on the precise spectral and geometric setting (Baskin et al., 2022).

Polyhomogeneous Expansions and Boundary Regimes

Solutions admit detailed polyhomogeneous expansions at different faces encoding the approach to timelike infinity, with decay exponents determined by the first resonant poles of the normal operator family. These expansions smoothly interpolate between decay rates at fixed $r$ and along outgoing null directions, elucidating the geometric interface of different temporal decay regimes (Baskin et al., 2022).

5. Domain-Specific Parameters and Physical Interpretation

The inverse square-root law is universally characterized by the exponent in the threshold singularity, with domain-specific parameters determining the amplitude and pre-exponential factors:

Domain	Spectral exponent $\alpha$	Observable	Temporal decay law
1D quantum decay / van Hove	$-1/2$	Survival amplitude $A(t)$	$t^{-1/2}$
Financial market impact (LLOB)	–	Impact propagator $G(\tau)$	$\tau^{-1/2}$
Wave equation, inverse-square pot.	(via spectral mapping)	Solution $u(t, r, \omega)$	$t^{-3/2}$, $t^{-1/2}$

• In quantum decay, $K = C\,\Gamma(1/2)\,\hbar^{1/2} e^{-i\pi/4}$ with $C$ determined by the residue expansion at threshold (Jiménez et al., 2018).
• In finance, $A$ and $s_0$ are empirically extracted from regression and relate to latent liquidity slopes (Maitrier et al., 22 Feb 2025).
• For wave equations, exponents reflect resonant geometric features and potential coupling strength (Baskin et al., 2022).

6. Critical Timescales, Crossovers, and Limitations

Both quantum and financial systems exhibit distinct regimes: initial short-time behavior, intermediate exponential or diffusive relaxation, and late-time power-law decay:

• Quantum decay: The transition from exponential to inverse-square-root decay occurs at a "critical time" $t_c \sim 2(\alpha+1)/\Gamma$, generally much later than accessible experimentally for narrow resonances (Jiménez et al., 2018, Jiménez et al., 2024).
• Market impact: The $\tau^{-1/2}$ regime persists for practical times ($\sim$ tens of minutes), saturating at larger lags due to external effects such as market adaptation or order-flow detection (Maitrier et al., 22 Feb 2025).
• Wave equations: Asymptotic exponents govern the decay beyond the light cone, with rates modified by the potential strength and spatial infinity geometry (Baskin et al., 2022).

Short-distance or short-time cutoffs (e.g., $s_0$ in price impact, finite system size or energy band limitations in quantum models) regularize singularities and determine the validity regime of the inverse square-root law.

7. Broader Implications and Comparisons

The emergence of the $t^{-1/2}$ temporal decay law across a range of physical, mathematical, and economic systems is a direct consequence of foundational structural properties—singularities at thresholds, diffusion of conserved quantities, or memory effects in collective modes. Its robustness and universal character contrast with context-dependent exponential laws or faster power-law decays. The law's emergence in empirical market data, quantum decay, and singular partial differential equations underscores its essential status in non-equilibrium relaxation phenomena (Jiménez et al., 2018, Jiménez et al., 2024, Maitrier et al., 22 Feb 2025, Baskin et al., 2022).

A plausible implication is that further studies of singular threshold behavior in other domains—or models incorporating diffusive relaxation and nontrivial spectral densities—can be expected to exhibit similar slow, universal temporal decay, modulated by system-specific amplitudes and crossovers.