Fisher Floor in Scaling and Information Theory

• Fisher Floor is defined as the minimal threshold in predictive information and scaling behavior, marking an irreducible lower bound in systems governed by Fisher-based metrics.
• It unifies disparate domains by identifying limits in fluctuation-response relations, logarithmic corrections, and statistical estimation, informing models in physics, finance, and networks.
• Applications range from resolving finite-size scaling discrepancies in critical phenomena to ensuring convergence in market and neural network algorithms, highlighting its universal relevance.

The Fisher Floor delineates the minimal threshold in predictive information or scaling behaviour that arises in systems where Fisher’s fluctuation-response relation, Fisher information, or Fisher zeros play a fundamental role. It is context-dependent but universally marks a lower bound set by the structure of Fisher-related quantities—whether as a limit in statistical estimation, a minimum sustained correlation in critical phenomena above upper critical dimensions, or as a convergence “floor” in iterative market processes. This concept appears in statistical field theory, market mechanisms, finance models, and information-theoretic neural network compression, always with technical roots in the interplay between Fisher-based bounds and the nature of physical, economic, or algorithmic systems.

1. Fisher’s Fluctuation–Response Relation and the Scaling Floor

Fisher’s fluctuation-response relation, expressed as $\eta = 2 - (\gamma/\nu)$, is a cornerstone in the scaling theory of critical phenomena. It connects the anomalous dimension $\eta$—which governs the spatial decay of the two-point correlation function %%%%2%%%%—with thermodynamic exponents $\gamma$ (susceptibility divergence) and $\nu$ (correlation length scaling).

Above the upper critical dimension ($d_c$), the scaling structure is profoundly altered. The physical correlation length $\xi_L$ no longer scales as $L$ but as $L^\varphi$ with $\varphi = d/d_c$. This adjustment, due to the presence of dangerous irrelevant variables (notably in the Landau-Ginzburg-Wilson action), induces a dichotomy:

• On the correlation-length scale, Fisher’s relation holds in its standard form with $\eta=0$ and susceptibility and correlation length exponents taking their mean-field values.
• On the system-length scale, a modified anomalous dimension $\eta_Q = \eta + 2(1-\varphi)$ appears, giving rise to observed negative values in finite-size simulations, for example $-1/2$ in the 5D Ising model.

This bifurcation determines a “Fisher floor”: a minimal, dimension-dependent correlation decay beyond which observables cannot fall, even when naïve scaling laws might suggest otherwise (Kenna et al., 2014).

2. Anomalous Dimension, Length Scale Dependence, and the Floor Phenomenon

The anomalous dimension $\eta$ quantifies deviations from canonical (mean-field) behaviour in fluctuation spectra. For $d \leq d_c$, finite-size scaling relates directly to the system size, so $\eta$ from simulations matches the field-theoretic prediction.

Above $d_c$, care must be taken to distinguish between the correlation-length scale (where $\eta=0$) and the system-length scale $L$, where a negative “effective” anomalous dimension arises. This distinction is formalized through “Q–FSS”, where the decay on the system scale is $G_Q(0, r) \sim r^{-(d-2+\eta_Q)}\,D_Q(r/L)$, with $\eta_Q$ as above. The relation $\eta_Q = \eta + 2(1-\varphi)$ ensures consistency, setting a quantifiable lower bound—the Fisher floor—on the measured anomalous dimension in finite-size realizations.

This phenomenon resolves the longstanding discrepancy between simulation and field-theoretic scaling above $d_c$, providing a unified understanding of lower bounds in the decay of critical correlations (Kenna et al., 2014).

3. Logarithmic–Correction Exponents at the Upper Critical Dimension

At $d = d_c$, dangerous irrelevant variables become marginal, and scaling forms acquire multiplicative logarithmic terms, e.g., $$\chi_\infty(t) \sim t^{-\gamma}|\ln t|^{\hat\gamma}$$, $\xi_\infty(t) \sim t^{-\nu}|\ln t|^{\hat\nu}$. Two Fisher-like relations for corrected exponents emerge:

• On the correlation-length scale: $\hat\gamma = (2-\eta)\hat\nu + \hat\eta$
• On the system-length scale: $$\hat\gamma = (2-\eta_Q)(\hat\nu - \hat\varphi) + \hat\eta_Q$$, with $\hat\eta_Q = \hat\eta + (2-\eta)\hat\varphi$

This bifurcation of logarithmic correction exponents constitutes a Fisher floor in the logarithmic regime: for example, in the 4D Ising model where $\hat\eta=0$, one finds numerically $\hat\eta_Q=1/2$ (Kenna et al., 2014).

This dual structure prescribes the minimal possible corrections to scaling observables and benchmarks simulation results against field-theoretic expectations, especially in the context of logarithmic deviations.

4. Implications in Percolation Theory and the Ginzburg Criterion

The Fisher floor extends decisively into percolation and fluctuation theory:

• In percolation above $d_c=6$, classical theory predicts $N_L \sim L^{d-d_c}$ divergent spanning cluster number. Distinguishing between system- and correlation-length scales reveals that $N_L$ is actually finite ($L^0$), resolving physical inconsistencies and aligning percolation scaling with hyperscaling when $\varphi$ is introduced.
• For the Ginzburg criterion, which sets the upper critical dimension via the comparison of fluctuation and mean-field contributions, the Fisher floor ensures that fluctuation decay on the system scale ($\eta_Q$) does not contradict mean-field expectations for exponents, but modifies the nature of observable fluctuations in finite systems (Kenna et al., 2014).

These analyses demonstrate that Fisher floors are not only mathematical limits but have direct relevance for physical predictions, cluster counting, and the interpretation of scaling breakdowns.

5. Fisher Information: Theoretical Lower Bound and “Floor” in Statistical Estimation and Dynamics

In statistics and information theory, the Fisher information $I = \int dx\, p(x)\, [d\ln p(x)/dx]^2$ sets a lower bound on the variance of unbiased estimators (Cramér–Rao inequality). In financial modeling, minimizing Fisher information under empirical constraints defines the minimal “information content” required for robust price prediction.

This minimum—the “Fisher Floor”—represents a sharp limit: regardless of model sophistication, there remains an irreducible uncertainty in prediction bounded from below by Fisher information. In quantum finance mappings, this floor underpins the “quantum-like” models for returns and asset prices, justifying the necessity of Gaussian, Laplacian, or power-law distributions depending on the information-theoretic structure and constraints (Nastasiuk, 2015).

6. Algorithmic and Computational Manifestations: Fisher Floor in Market and Network Models

In iterative market mechanisms such as tâtonnement in linear Fisher markets, a “Fisher floor” manifests as a strict lower bound on iterates and convergence radii:

• The step size $\eta$ in subgradient descent governs not only rate of convergence but also proximity to equilibrium—a residual error floor $e$ arises, vanishing only as $\eta \to 0$.
• Initial prices must be set strictly above a threshold vector (“floor”) defined by budgets, valuations, and item number to guarantee bounded positive iterates and stability.
• Numerical experiments confirm that this error floor is unavoidable, functionally dictated by Fisher-type quantities in convex program formulations (Nan et al., 18 Jun 2024).

Similarly, in neural network compression, the full Fisher information matrix enables quantification of parameter sensitivity. Compression algorithms that only use diagonal approximations ignore parameter correlations, potentially pushing models below their Fisher floor and losing essential information. Advanced methods such as Generalized Fisher-Weighted SVD (GFWSVD) approximate the full Fisher information via Kronecker-factoring, thus adhering to a more accurate Fisher floor and preventing over-compression that sacrifices performance (Chekalina et al., 23 May 2025).

7. Fisher Zeros and Quantum Criticality: The Analytical “Floor” in Complex Partition Functions

In quantum systems, Fisher zeros—the roots of the analytically continued partition function $Z(\beta)$—enable the identification of phase transitions without singularities on the real axis. The distribution of Fisher zeros forms open or closed curves in the complex $\beta$ plane with fundamental changes (such as open lines disappearing at a quantum critical point) marking a critical Fisher floor for the dynamics and thermodynamics (Liu et al., 27 Jun 2024).

This analytical structure determines not only equilibrium properties but also sets minimal bounds for dynamical observables such as survival amplitudes in thermofield double states, further confirming the role of the Fisher floor as a universal bound across statistical, quantum, and information-theoretic disciplines.

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The Fisher Floor thus constitutes a rigorous limitation governed by Fisher’s scaling relations, information bounds, and structural zeros in analytic continuations. Its consistent appearance across disparate domains—critical phenomena, finance, market algorithms, neural networks, and quantum systems—emphasizes its foundational role as the definitive lower bound in the encoding, propagation, and estimation of information or correlation in complex systems.