Dilution Effect in Complex Systems

• Dilution effect is the reduction of a system’s key properties, such as magnetic ordering or detection sensitivity, by introducing inactive or foreign elements.
• Studies employ empirical, computational, and theoretical methods to quantify changes in phenomena like Néel temperature reduction, percolative conduction, and signal dilution.
• Applications include tuning magnetic orders, engineering electronic phases, and enhancing methodologies in group testing, cosmology, and astrophysical observations.

The dilution effect refers to the reduction of a relevant physical, biological, or observational property—such as magnetic ordering temperature, metallicity, detection sensitivity, acoustic loss, or disease risk—due to the introduction of non-participating constituents, disorder, or increased system complexity. In all domains, this effect quantifies how the presence of "inactive," "non-magnetic," "non-informative," or "foreign" elements suppresses collective or local phenomena. The specifics of the dilution mechanism depend on the underlying interactions, dimensionality, disorder, and the nature of the observable. Empirical, computational, and theoretical studies have provided quantitative frameworks for describing the dilution effect in strongly-correlated magnets, topological phases, group testing, cosmology, signal detection, condensed matter, and ecology.

1. Magnetic Dilution in Quantum Magnets

Non-magnetic ion substitution disrupts exchange paths in spin lattices, leading to pronounced suppression of magnetic long-range order and crossover to disordered regimes. In the frustrated 3D antiferromagnet Li$_2$CuW$_2$O$_8$, Zn$^{2+}$ substitution on the Cu$^{2+}$ sites reduces the Néel temperature $T_{\rm N}$ exponentially with Zn fraction $x$: $T_{\rm N}(x) = T_{\rm N}(0)\, e^{-8x}$ with $T_{\rm N}(0) \simeq 3.9\,\mathrm{K}$ and $T_{\rm N}(0.25) \simeq 0.53\,\mathrm{K}$ (Ranjith et al., 2014). The exponential law is emblematic of quasi-one-dimensional antiferromagnets but here occurs in a 3D frustrated network due to the amplification of low-dimensional fluctuations by geometric frustration. The low-temperature heat capacity transitions from a 3D magnon ($_2$0) in the parent compound to nearly linear ($_2$1) under strong dilution. The dipolar nature of Li–Cu coupling and the value of the maximum exchange constant ($_2$2) remain robust under Zn-doping. Similar effects are observed in topological antiferromagnets such as (Mn$_2$3Pb$_2$4)Bi$_2$5Te$_2$6, where Pb substitution linearly suppresses the Néel temperature and both interlayer exchange and anisotropy scales, allowing controlled tuning through magnetic, structural, and topological regimes (Qian et al., 2022).

2. Dilution in Electronic Transport and Correlated Phases

Site dilution—removal or replacement of interacting sites by non-participating atoms—can drive dramatic changes in electronic and magnetic ground states of correlated materials. In SrRu$_2$7Ti$_2$8O$_2$9 and SrRu$_8$0Ga$_8$1O$_8$2, nonmagnetic substitutions (Ti$_8$3, Ga$_8$4) decrease the macroscopic ordered moment, suppress or fragment magnetic order (rising critical exponent $_8$5), and induce Griffiths-phase–like regimes above $_8$6 (Gupta et al., 2017, Gupta et al., 2019). In the case of the half-filled, diluted Hubbard model with long-range hopping (Mandal et al., 30 Jun 2025), setting $_8$7 at a fraction $_8$8 of lattice sites creates impurity bands, enabling percolative conduction without destroying long-range antiferromagnetic (AF) order. The critical dilution $_8$9 at which the Mott gap closes and metallicity arises can be orders-of-magnitude reduced by long-range hopping $^{2+}$0, and further engineering of sublattice-dependent hopping enables half-metallic AF phases: $^{2+}$1 where only one spin channel remains gapless.

3. Dilution Effect in Group Testing, Epidemiology, and Signal Detection

The "dilution effect" also quantifies the loss in detection sensitivity when signals (biological or physical) are mixed with background or uninformative components. In group testing for viral prevalence, sample pooling leads to reduced analyte concentration per test, and hence a concentration-dependent (dilution) loss: $^{2+}$2 where $^{2+}$3 models the PCR sensitivity as a strictly increasing function of concentration (Wan et al., 2021). The false-negative rate for Dorfman pooling rises with pool size (hence dilution). Correlated pooling partially mitigates this, but sensitivity and efficiency trade-offs are fundamentally constrained by the dilution curve $^{2+}$4.

In wireless sensing, the dilution effect occurs when uninformative links—whose Fresnel zones do not intersect activity regions—are aggregated, thereby overwhelming informative signals with noise. A single well-placed link ($^{2+}$5) outperforms the full mesh ($^{2+}$6), with monotonic decline in performance as more non-informative links are included (Rodrigues et al., 11 Feb 2026).

4. Dilution Effects in Cosmology and Astroparticle Physics

Entropic dilution in cosmology pertains to the reduction of a frozen-out dark matter (DM) relic density through entropy injection during a first-order phase transition. During such a transition, if the DM freeze-out temperature $^{2+}$7 precedes the nucleation temperature $^{2+}$8, then

$^{2+}$9

with dilution factor $^{2+}$0 (2207.14519). However, in models like the xSM or NMSSM, the condition $^{2+}$1 is always satisfied, so entropy injection leaves $^{2+}$2 unaffected; only in models where $^{2+}$3 (e.g., heavy singlet DM in 2HDM+S) is significant dilution ($^{2+}$4) realized.

A closely related concept, the "dilution-resistant effect," arises for light species produced from neutrinos after decoupling. Energy stored in massive states dilutes more slowly than the radiation energy, leading to a minimal, robust contribution to $^{2+}$5 in the cosmic background: $^{2+}$6 for scalar and vector mediators, respectively (Li et al., 2023).

5. Dilution of Observable Properties in Astrophysics and Weak Lensing

In photometric cosmology, the dilution effect manifests as the lowering of cluster weak-lensing signals due to the inclusion of foreground or member galaxies in the source catalog. These galaxies exhibit no coherent shear from the lens cluster but contribute to the noise. In the weak-lensing catalogs built from HSC survey data, adoption of a globally normalized estimator and photometric-redshift cuts efficiently suppresses dilution:

• Cluster-member dilution is eliminated by global denominator normalization.
• Foreground dilution is mitigated by excluding low-$^{2+}$7 sources using "P-cut" criteria on full photo-z PDFs.
• Combining multiple photo-z cuts recovers nearly double the number of secure cluster detections at $^{2+}$8 (Hamana et al., 2020).

6. Dilution in Galaxy Metallicity and Merger-Driven Evolution

In galaxy evolution, "dilution" refers to the reduction of central gas-phase metallicity due to the inflow of metal-poor gas, particularly during or after mergers:

• In semi-analytic models and cosmological simulations, minor merger–driven accretion of low-$^{2+}$9 gas reduces $T_{\rm N}$0 on Gyr timescales after a major merger quenches star formation (Yates et al., 2013). This creates a characteristic positive slope in the high-mass end of the fundamental metallicity relation ($T_{\rm N}$1).
• In simulated major mergers, merger-induced metallicity dips are typically $T_{\rm N}$2 dex for $T_{\rm N}$3 (mass ratio), and $T_{\rm N}$4 dex for $T_{\rm N}$5 (minor), localized to $T_{\rm N}$6 kpc. Mergers systematically populate the low-metallicity outlier tail of the FMR residuals (Bustamante et al., 2017).

7. Statistical, Ecological, and Theoretical Dilution Phenomena

In ecological epidemiology, the "dilution effect" denotes the widely observed reduction of mean disease risk, as well as variance of epidemic severity, with increasing host species richness. Mathematically, frequency-dependent transmission models exhibit: $T_{\rm N}$7 where $T_{\rm N}$8 is the number of host species; both mean and variance of the community reproductive number shrink, yielding safer and more predictable outcomes. For density-dependent transmissions, this effect is absent and may even be reversed (Shaffery et al., 2019).

In statistical mechanics, for the mean-field Ising model with dilution fraction $T_{\rm N}$9, the effective coupling is rescaled $x$0, shifting spinodal and critical loci but leaving exponents unchanged as long as the Ginzburg parameter $x$1. In finite-range systems, dilution reduces the Ginzburg parameter and thus broadens the susceptibility peak (pseudospinodal), with rounding width: $x$2 (Liu et al., 2013). Below the upper critical dimension, site dilution becomes relevant in the Harris sense and can alter universality.

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These diverse manifestations of the dilution effect demonstrate its central role across multiple disciplines—quantifying the suppression or modification of collective, structural, dynamical, or observational properties due to the admixture of inactive, non-informative, or extraneous elements. The precise functional dependence of dilution-induced suppression encodes rich information about system connectivity, frustration, dimensionality, mode-structure, and the statistics of disorder.