Fisher Threshold Theorem Overview

Updated 8 October 2025

Fisher Threshold Theorem is a framework identifying critical parameters where qualitative transitions occur in diverse systems such as probability, physics, and genetics.
It quantifies thresholds in limit theorems via Fisher information decay, spectral gap analysis, and cumulant vanishings in i.i.d. sum models.
The theorem also delineates error thresholds in population genetics, phase transitions in statistical physics, and optimal bounds in combinatorial designs and signal detection.

The Fisher Threshold Theorem encapsulates a collection of threshold phenomena in probability, statistical physics, combinatorics, information theory, and population genetics, each emerging from the interplay between Fisher’s metrics (e.g., Fisher information, partition function zeros, genetic variance, combinatorial intersection bounds) and structural or dynamical constraints on the underlying systems. The term is used to denote the existence of critical parameters or rates at which qualitative changes occur in convergence, phase transitions, population stability, combinatorial design bounds, or signal detection. Across mathematical biology, probability theory, combinatorics, and quantum/statistical physics, this threshold governs transitions in system behavior, criticality, and optimality.

1. Fisher Thresholds in Probability and Information Theory

Central to statistical limit theorems, the Fisher Threshold Theorem characterizes how Fisher information decays in sum sequences of independent (or nonlinear) random variables. Let $X_1,\dots,X_n$ be i.i.d. with mean zero and variance $\sigma^2$ , and let $Z_n = (X_1+\cdots+X_n)/\sqrt{n}$ . The relative Fisher information is

$I(Z_n\|\mathcal{N}) = I(Z_n) - I(\mathcal{N}),$

where $\mathcal{N}$ denotes the normal law. Via Edgeworth-type expansions, it is established that

$I(Z_n\|\mathcal{N}) = \frac{c_1}{n} + \frac{c_2}{n^2} + \cdots + o\left(n^{-(s-2)/2}(\log n)^{(s-3)/2}\right),$

with $c_j$ determined by the cumulants of $X_1$ beyond order two (Bobkov et al., 2012). If all lower cumulants vanish, the leading coefficients are zero until the first nonzero $c_j$ (“Fisher threshold” phenomenon).

Further, via spectral analysis of the convolution operator and maximal correlation (Hirschfeld-Gebelein-Rényi), the standardized Fisher information $J_{st}(Z_n)$ obeys

$J_{st}(Z_n) \leq \frac{J_{st}(X_1)}{1+\varepsilon(2)(n-1)},$

where the spectral gap $\varepsilon(2)$ linked to the second-largest eigenvalue controls the $O(1/n)$ convergence rate and monotonicity (Johnson, 2019). This establishes a quantitative threshold for Gaussianization in information-theoretic CLTs.

Recent work extends decomposition techniques (martingale or approximation by independent components) to nonlinear statistics $F$ , yielding new bounds:

$I(F\|\mathcal{N}) \leq 2\,\mathbb{E}\left[\left(\frac{1-O_F}{O_F}\right)^2\right] + 2\,\mathbb{E}\left[\left(\frac{O_{F,F}}{O_F}\right)^2\right],$

providing explicit $O(1/n)$ rates for quadratic forms and functions of sample means, generalizing the Fisher threshold to broader settings (Dung, 2022).

Setting	Threshold Parameter	Rate/Transition
i.i.d. sums CLT	moment-based cumulant vanishings	faster than $1/n$
Maximal correlation spectral gap	2nd-largest eigenvalue $\varepsilon(2)$	$O(1/n)$ bound
Nonlinear statistics	deviation of score from linearity	info-threshold

2. Fisher Zeros and Thresholds in Statistical Physics

In the theory of phase transitions, the Fisher Threshold Theorem refers to the locus of Fisher zeros—roots of the partition function in a complexified temperature (or coupling) parameter—approaching the real axis, thereby revealing thermodynamic or quantum critical points (Liu et al., 27 Jun 2024, Liu et al., 2018). For the one-dimensional transverse field Ising model (1DTFIM), the partition function

$Z(\beta, g) = \frac{1}{2}\left[\prod_{k=1}^L 2\cosh(\beta \epsilon_{2k}) + \text{sign}(g_c - g)\prod_{k=1}^L 2\sinh(\beta \epsilon_{2k}) + \ldots\right]$

has zeros (Fisher zeros) which, for large $L$ , form open or closed lines in the complex $\beta$ plane. The topology of these lines changes at the quantum critical point $g = g_c$ :

For $g < 1$ : open lines associated with ferromagnetic domain walls
At $g = g_c$ : open lines vanish, closed loops persist (disordered phase)
Fishers zeros serve as dynamical (quantum) or thermodynamic phase transition indicators

Algorithmic studies of the Ising model on graphs of bounded degree reveal that, in the correlation decay (uniqueness) region for $\beta$ , the partition function is analytic, and Fisher zeros are absent in a complex neighborhood (Liu et al., 2018). This absence signals no phase transition and enables efficient deterministic approximations.

Model/Class	Fisher Threshold	Physical Transition
1DTFIM and spin models	Topology of Fisher zeros	quantum/thermal criticality
General Ising graphs	Absence of zeros in region	analyticity, no transition

3. Thresholds in Population Genetics

In evolutionary dynamics, the Fisher Threshold Theorem denotes the conditions under which genetic information (e.g., high-fitness “master sequences”) is maintained in the face of stochastic effects (mutation, drift). In the finite population Wright-Fisher model (Cerf, 2012), the critical threshold is expressed as

$\alpha\,\psi(a) = \ln \kappa$

where $\alpha = m/\ell$ (scaled population size), $\psi(a)$ (cost function via Freidlin-Wentzell large deviations), and $\kappa$ (alphabet size) demarcate regimes:

$\alpha\,\psi(a) < \ln\kappa$ : error catastrophe, random population
$\alpha\,\psi(a) > \ln\kappa$ : stability, quasispecies formation

In classical infinite-population quasispecies (Eigen model), the error threshold appears as $\sigma e^{-a} = 1$ . These results rigorously connect deterministic threshold phenomena to stochastic finite-population effects through large deviation theory.

4. Fisher Thresholds in Combinatorics and Design Theory

The combinatorial Fisher Threshold Theorem generalizes Fisher’s inequality to set systems and its q-analogues (Dey, 21 May 2025, Das et al., 2014). For sets (or subspaces) with prescribed intersection properties:

Classical: any family $\mathcal{F}$ on $[n]$ with $|\mathcal{F}|$ subsets such that $|A \cap B| = k$ for any $A \neq B$ in $\mathcal{F}$ satisfies $|\mathcal{F}| \leq n$
$q$ -analogue: for $\mathcal{F}$ a family of subspaces of $\mathbb{F}_q^n$ with $dim(A \cap B) = k$ , $|\mathcal{F}| \leq [n]_q$

Relaxing intersection conditions (allowing up to $k$ exceptions per set) increases the maximum attainable family size:

For $k$ -almost $\lambda$ -Fisher families, bounds interpolate between $n$ and $2n-2$ with the “Hadamard construction” optimal when $\lambda = n/4$ (Das et al., 2014)

Setting	Threshold Parameter(s)	Max Family Size Bound
Sets, $\|A\cap B\|=k$	$n$	$\|\mathcal{F}\|\leq n$
$q$ -analogues	$q$ , $n$	$\|\mathcal{F}\| \leq [n]_q$
$k$ -almost Fisher families	$k, n, \lambda$	Linear in $k$ , tight @ Hadamard

5. Thresholds in Signal Detection and Meta-Analysis

Statistical thresholding appears in the optimal design of $p$ -value combination methods. The TFisher framework generalizes Fisher’s method by truncating and weighting $p$ -values (Zhang et al., 2018).

For sparse signals, hard or soft thresholding (e.g., inclusion of only $p$ -values below a data-adaptive cutoff $\tau$ ) is optimal
The Bahadur Efficiency (BE) and Asymptotic Power Efficiency (APE) metrics reveal that soft-thresholding ( $\tau_1 = \tau_2$ ) often achieves maximal power
Adaptive thresholding, as in the omnibus oTFisher test, achieves near-optimal detection even without prior signal proportion knowledge

Approach	Threshold Mechanism	Signal Detection Context
Fisher combination	none (all $p$ -values)	dense signal, less selective
TFisher soft/hard	adaptive cutoff $\tau$	sparse signal, optimal power
oTFisher (omnibus)	grid of thresholds	robust across regimes

6. Thresholds in Evolution and Dynamical Systems

The original Fisher’s Fundamental Theorem (as interpreted in (Frank, 2011)) partitions evolutionary change in mean fitness as

$\Delta W = \Delta W_{NS} + \Delta W_E,$

where $\Delta W_{NS}$ (due to natural selection) equals the genetic variance in fitness, and $\Delta W_E$ captures environmental deterioration. The “Fisher threshold” corresponds to the dynamic equilibrium where improvement via selection is counterbalanced by environmental or competitive effects, leading to near-constancy of mean population fitness over time.

This invariance law is juxtaposed with Wright’s dynamical models, revealing a conceptual threshold between universal conservation principles and context-dependent evolutionary trajectories.

7. Thresholds Characterizing Criticality and Phase Transitions

Across disciplines, the Fisher threshold serves as a diagnostic or bifurcation point, signifying transitions in system behavior:

Convergence to Gaussianity (probability/information theory)
Emergence or disappearance of genetic information (population biology)
Existence or absence of phase transitions (quantum/statistical physics)
Maximal configuration numbers under combinatorial constraints (design theory)
Statistical power optimality under signal sparsity (meta-analysis)

These threshold phenomena are unified by the role of Fisher’s conceptual or quantitative metrics in structuring the space of possible behaviors, often expressed as sharp inequalities, spectral parameters, or topological transitions in zeros or maxima.

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