Achievable Burning Densities in Propagation Models

• Achievable burning densities are the limiting fractions of activated sites in dynamical systems, quantifying propagation in graphs, hypergraphs, and fusion models.
• Analytical methods use discrete-time dynamics, scaling laws, and threshold criteria to establish phase transitions and establish maximal burning coverage.
• Insights into burning thresholds guide practical applications in fusion plasma optimization and astrophysical phenomena by outlining critical propagation regimes.

Achievable burning density is a concept quantifying the asymptotic fraction of “burned” sites (e.g., activated, affected, or ignited) in a growing medium under a specified propagation mechanism. It arises in contexts ranging from discrete dynamical models of contagion or fire in networks, to the attainment of fuel densities in fusion systems, and to threshold phenomena in hypergraph-based influence dynamics. The term formalizes, for a given burning protocol and scaling of the underlying structure, the limiting proportion of the system reached by the “burn” process, and characterizes both maximal and minimal sustained coverage under optimal or constrained strategies.

1. Burning Processes: Definitions and Formal Setting

A burning process is typically modeled as a discrete-time dynamical system on a sequence of (growing) graphs or hypergraphs, where in each round previously burned vertices propagate the burn to their neighbors, and an “activator” vertex may be additionally set on fire. For a growing grid $G_n$, an activator sequence $\{v_n\}$, and burned set $B_n$ after $n$ steps, the burning density is defined as

$\delta = \lim_{n\to\infty} \frac{|B_n|}{|V(G_n)|}$

whenever the limit exists. For fixed or random activator protocols, the set of achievable burning densities $\mathcal{P}$ is the range of densities obtainable for some sequence, and is central to the phase diagram of the system (Barrett et al., 20 Jan 2026, Bonato et al., 2018).

In proportion-based hypergraph burning, a round-indexed procedure proceeds by activating vertices in $V(H)$ and propagating fire across hyperedges $e$ once a critical proportion $p|e|$ of their vertices are already on fire. The burning distribution $f_H(p)=b_p(H)$ partitions the domain of $p$ into intervals where the burning time remains constant, and these intervals directly encode the “achievable” burning densities for each propagation regime (Burgess et al., 2024).

2. Achievable Burning Densities in Growing Grids

The archetypal setting is burning on $\mathbb{Z}^2$ grids growing as $G_n = [-f(n),f(n)]^2$ with $f(n) = \lceil c n^\alpha \rceil$, $c > 0$, $\alpha \geq 1$. The main theorem gives a sharp characterization:

• For $\alpha=1$, $\mathcal{P} = [1/(2c^2), 1]$.
• For $1 < \alpha < 3/2$, $\mathcal{P} = [0,1]$.
• For $\alpha = 3/2$, $\mathcal{P} = [0,(1+\sqrt{6}c)^{-2}]$.
• For $\alpha > 3/2$, $\mathcal{P} = \{0\}$ (Barrett et al., 20 Jan 2026).

This exhibits two critical exponents $\alpha=1$ and $\alpha=3/2$:

• For $\alpha < 1$, the grid grows sub-linearly; the burn process eventually covers all vertices.
• For $\alpha = 1$, only densities above a universal minimum are possible, tied to the fire's linear expansion front.
• For $1 < \alpha < 3/2$, fire outpaces grid growth, allowing any fractional coverage up to full burning.
• For $\alpha = 3/2$, the fire “competition” is balanced, restricting the maximal density to $(1+\sqrt{6}c)^{-2}$.
• For $\alpha > 3/2$, grid area $n^{2\alpha}$ outgrows any $O(n^3)$ expansion attainable by the burning process, forcing limiting density zero (Barrett et al., 20 Jan 2026, Bonato et al., 2018).

The critical regime $\alpha=3/2$ is controlled by explicit spatial–temporal tiling and recursive activation strategies, proving tight upper and lower bounds for the maximal density. For higher-dimensional lattices, the exponent generalizes to $\alpha = (d+1)/d$ for grids in $\mathbb{Z}^d$ (Bonato et al., 2018).

3. Achievable Densities in Hypergraph Burning

In hypergraph settings, the parameter space of achievable burning densities and times is governed by both the structure of the hypergraph (edge sizes, degree, automorphism group) and the propagation parameter $p$ (Burgess et al., 2024). For any $k$-uniform connected $H$, the burning distribution decomposes $(0,1)$ into intervals: $$(0,1) = \bigsqcup_{k=1}^{|V|} P_k, \quad P_k = \{p: b_p(H) = k\}$$ with each $b_p(H)$ constant on intervals $(m/k,(m+1)/k]$, $m=0,1,...,k-1$. Each such interval corresponds to an achievable burning “rate” or density.

Lower and upper bounds link the achievable burning time to extremal properties of the hypergraph's edges: $$\min_{e\in E} \lceil p|e|\rceil \leq b^{L}_p(H) < b_p(H) \leq 1 + \sum_{e\in E}\lceil p|e|\rceil$$ with tightness realized in extremal constructions. For balanced incomplete block designs, the order of the automorphism group empirically correlates with the maximal lazy burning number, reflecting that increased symmetry can impede burning (Burgess et al., 2024).

4. Burning Density Thresholds in Fusion Plasmas

In magnetically confined fusion, achievable burning density is controlled by the balance between heating power, radiation losses (especially from impurities), and particle/energy confinement. The classical Greenwald limit $n_G = I_p/(πa^2)$ is supplemented by models such as plasma-wall self-organization (PWSO), which introduce heating-power-dependent limits (Liu et al., 19 Feb 2025, Liu et al., 5 May 2025):

• At fixed plasma size and auxiliary heating, PWSO yields $n_\text{lim} \propto P^\alpha$ with $\alpha \approx 0.25-0.75$, depending on divertor/edge conditions and impurity yields.
• Experiments on EAST with ECRH-assisted Ohmic startup achieve $n_e$ up to $1.65\,n_G$, reaching the predicted "density-free" regime at low divertor target temperatures $T_t<7$ eV (Liu et al., 5 May 2025).
• In high power and optimized edge configurations, central densities $n_e \sim 1$–$2\times10^{20}\,{\rm m}^{-3}$ are attainable, approaching or exceeding the threshold for self-sustaining burning plasma operation (Liu et al., 19 Feb 2025).

Relevant density thresholds are:

• For regime transition (e.g., self-heating to propagating burn in ICF): $\rho_\text{peak} \gtrsim 25\,{\rm g/cm}^3$ and $\rho R \gtrsim 0.75\,{\rm g/cm}^2$ (Tong et al., 2019, Ross et al., 2021).

5. Burn-Up Thresholds in Spherical Thermonuclear Fuels

In spherical ignition of dense fuels (e.g., CD$_4$, CD$_2$T$_2$ methane), the critical burn-up parameter $x_c = \rho_0 r_f$ sets the minimal areal density required for self-sustained detonation (Frolov, 2010):

• For $T=5$–20 keV, CD$_4$ requires $\rho_0 \gtrsim 5\times10^3\,{\rm g/cm}^3$ for ignition.
• CD$_2$T$_2$ ignites at $\rho_0 \sim 70$–$100\,{\rm g/cm}^3$, supporting burning at densities within the range of DT ICF devices.

These thresholds are set by the coupling between energy release, heat transport, and hydrodynamics, and are sensitive to the initial composition, temperature, and symmetry of the system (Frolov, 2010).

6. Astrophysical and Cross-Disciplinary Manifestations

In stellar environments, burning density may describe the fraction of fuel consumed or the influence of non-standard energy sources (e.g., WIMP dark matter annihilation) on observable properties. For canonical WIMPs (mass $m_\chi=100$ GeV, cross-section $\sigma_\text{SD}=10^{-38}\,{\rm cm}^2$), burning effects in a cluster require local DM densities $\rho_\chi \gtrsim 10^9-10^{10}\,{\rm GeV\,cm}^{-3}$; for $m_\chi=8$ GeV, the required density drops to $\sim 3\times10^5\,{\rm GeV\,cm}^{-3}$ (Casanellas et al., 2011). Only in such high-density environments do dark-matter-fueled burning alter the stellar “density” of burned matter (i.e., energy output per unit mass) enough to affect observables.

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The study of achievable burning densities thus formalizes the interplay among the process kinetics (burn rules), the geometry and growth of the underlying structure, and the available energetic or environmental resources. It provides a critical lens for understanding phase transitions and maximal reach in discrete dynamical, physical, and astrophysical contexts (Barrett et al., 20 Jan 2026, Bonato et al., 2018, Burgess et al., 2024, Liu et al., 19 Feb 2025, Tong et al., 2019).