Critical Density for Activated Random Walk

• The paper establishes that μc is a sharp threshold distinguishing absorbing states from persistent activity in ARW, with rigorous bounds across various graph structures.
• It employs combinatorial techniques, stabilization via odometers, and multi-scale renormalization to determine how μc depends on sleep rate and jump distribution.
• The findings provide deep insights into phase transitions, self-organized criticality, and dynamical properties in non-equilibrium systems.

Activated Random Walk (ARW) features a non-equilibrium phase transition between fixation (absorbing state) and sustained activity, controlled by a critical particle density. The critical density for ARW is the sharp threshold distinguishing almost-sure absorption from a regime where activity persists indefinitely: when the initial density of particles exceeds this critical value, the system supports a stationary active phase, while below it the system fixates.

1. Definition and Formal Characterization of Critical Density

In ARW, each site of a graph (typically ℤᵈ or a vertex-transitive graph) hosts particles that are either active (A) or sleeping (S). Active particles perform continuous-time random walks; a solitary active particle may fall asleep at rate λ and reactivate if another active particle arrives. The critical density, denoted as μ_c = μ_c(λ, p(·)), is the infimum density (over product measures) for which there is a positive probability of sustained activity:

$$μ_{c}(λ, p(\cdot)) = \inf\left\lbrace μ : \mathbb{P}_μ(\text{system is active}) > 0 \right\rbrace.$$

Here, p(·) denotes the jump distribution, and λ is the sleep rate. For μ < μ_c, the system fixates (all sites eventually reach absorbing configurations); for μ > μ_c, activity persists with positive probability (Taggi, 2014).

2. Rigorous Bounds and Dependence on Model Parameters

The value of μ_c depends sensitively on the jump distribution, sleep rate, and underlying geometry:

• Biased Jumps (Nonzero Drift):
  • In one dimension:

  %%%%1%%%%

  and ℓ^{\mathcal{H}} is the time spent by the random walk in the half-space opposing the bias. - In higher dimensions:

  $$μ_{c}(λ, p(\cdot)) \leq \frac{1}{F(λ, p(\cdot)) + 1}$$ - The critical density strictly depends on the form and bias of p(·), and for systems with maximal drift, μ_c equals λ/(1+λ) (Taggi, 2014).

• Unbiased Jumps and Dimension Dependence:
  • On ℤ, μ_c < 1 when λ is sufficiently small, and μ_c(λ) → 0 as λ → 0 (Basu et al., 2015).
  • On ℤᵈ with d ≥ 3 or any transitive, transient graph, μ_c < 1 for all sleep rates, confirming that the system supports activity at densities strictly below full occupancy (Taggi, 2017).
  • On ℤ², for small λ:

  $ζ_c(λ) \leq \frac{C}{\ln(1 / λ)}$

  showing a logarithmic dependence on the inverse sleep rate (Hu, 2022). - On regular trees and general vertex-transitive amenable graphs, μ_c is strictly between 0 and 1, and a sharp lower bound μ_c ≥ λ/(1+λ) holds (Stauffer et al., 2015).

3. Mechanisms and Methods for Determining μ_c

The determination of μ_c employs a mix of combinatorial, probabilistic, and multi-scale renormalization techniques:

• Diaconis–Fulton Representation: The ARW is encoded via random stacks of instructions (moves or sleep) at each site; the abelian (commutative) property is leveraged so that the order of legal "topplings" does not affect stabilization outcomes (Taggi, 2014, Forien et al., 2022).
• Stabilization and Odometers: The odometer function records the number of topplings at each site during stabilization. Growth of odometer values on the order of system size implies sustained activity; bounded odometer values indicate fixation (Taggi, 2014).
• Recursive, Multi-scale Bounds: Probabilistic renormalization arguments track how local escape probabilities decay or persist under box-coarse graining, often requiring "sprinkling" (small density reductions) and buffer zones to decouple events across scales (Sidoravicius et al., 2014).
• Mass-Transport and Weak Stabilization: On vertex-transitive graphs and in the presence of heterogeneous site capacities, mass-transport arguments and "weak" stabilization principles yield sharp lower and upper bounds for μ_c (Chiarini et al., 2021, Stauffer et al., 2015).

4. Model Variants, Geometry, and Universality

The phase transition and the value of μ_c persist across diverse graph structures and model variants:

• Transitive, Transient, and Non-amenable Graphs: On transitive, transient graphs, μ_c → 0 as λ → 0. For regular trees of degree at least 3, μ_c ∈ (0,1) for all λ (Stauffer et al., 2015, Taggi, 2017).
• Mean-Field and High Dimensions: In the mean-field regime (complete graph or d → ∞), μ_c approaches the sleep probability:

$$μ_c(\mathbb{Z}^d, λ) = \frac{\lambda}{1+\lambda} + \frac{\lambda}{1+\lambda} \frac{1}{1+\lambda} \frac{1}{2d} + O\left(\frac{1}{d^2}\right)$$

with corrections reflecting the residual spatial return probability (Járai et al., 2023, Junge et al., 12 Sep 2025).

• Comb and Non-Euclidean Geometries: On the comb graph, both rigorous bounds and simulations show lower critical densities on the spine and teeth compared to the interval, highlighting sensitivity to local geometry (Junge et al., 20 Aug 2025).

5. Recent Progress: Density Conjecture and Exponential Tail Bounds

Recent works have provided a complete proof in dimension one of the "density conjecture": the stationary density attained under driven-dissipative dynamics coincides exactly with the critical density of the conservative (fixed-energy) ARW (Hoffman et al., 3 Jun 2024, Forien, 4 Feb 2025). Specifically, in one dimension and for each natural model variant (conservative, driven-dissipative, point-source, and cyclic), the critical threshold is the same. Furthermore, deviations from the critical density in the stationary state exhibit exponential decay in system size, both above and below the threshold (Hoffman et al., 3 Jun 2024). Superadditivity arguments, combined with layer-percolation methods, underlie these results and yield strong concentration around μ_c.

6. Implications and Applications

The existence and nontriviality of μ_c in ARW and related models have multiple implications:

• Self-Organized Criticality: ARW self-organizes to the critical state in the driven-dissipative setting, aligning with theoretical frameworks for self-organized criticality (SOC) (Hoffman et al., 3 Jun 2024).
• Sharp Phase Transition: The phase transition at μ_c is sharp: for densities just below, the system fixates with high probability; for densities just above, with high probability, the system stays active (Hoffman et al., 3 Jun 2024, Brown et al., 12 Nov 2024).
• Robustness and Universality: The critical density and phase transition are robust to changes in initial condition (e.g., activating only a single initially active particle among sleeping ones still yields the same threshold) (Brown et al., 12 Nov 2024).
• Mixing and Cutoff: The time needed to reach equilibrium (mixing time) in driven ARW on finite intervals exhibits cutoff at time nμ_c, emphasizing the dynamical significance of μ_c beyond mere static equilibrium properties (Hoffman et al., 29 Jan 2025).
• Capacity and Disorder: ARW with site-dependent (possibly random) capacities still exhibits a phase transition, with the critical threshold controlled by the mean capacity (Chiarini et al., 2021).

7. Summary Table: Selected Bounds for μ_c

Setting	μ_c Upper Bound	μ_c Lower Bound
Biased jumps on ℤ	1 − F(λ, p(·))	λ/(1+λ) (in special cases)
Symmetric jumps on ℤ (small λ)	C√λ	c√λ
ℤ² (small λ)	C / ln(1/λ)	—
Regular tree (deg ≥ 3)	< 1 (explicit bounds)	> 0
ℤᵈ, d → ∞	λ/(1+λ) + O(1/d)	λ/(1+λ)
Vertex-transitive amenable	< 1 for small λ	≥ λ/(1+λ)
Complete graph (mean-field)	λ/(1+λ)	λ/(1+λ)

*Constants C, c depend on the specific model and may be derived via coupling, percolation, or spectral estimates (Taggi, 2014, Stauffer et al., 2015, Asselah et al., 2019, Taggi, 2017, Járai et al., 2023, Junge et al., 12 Sep 2025, Hu, 2022).

8. Open Problems and Research Directions

Key directions include:

• Extending sharp characterization and concentration bounds for μ_c to higher dimensions and more general graphs (Stauffer et al., 2015, Taggi, 2017, Junge et al., 12 Sep 2025).
• Determining exact values and scaling laws for μ_c beyond the high-dimensional or mean-field limit, particularly in low-dimensional, recurrent geometries.
• Understanding universality classes and the effect of local structure (e.g., comb, trees, random environments) on μ_c (Junge et al., 20 Aug 2025).
• Developing quantitative and algorithmic approaches to estimating μ_c in models with heterogeneous or random site capacities (Chiarini et al., 2021).

9. Concluding Remarks

Research to date positions the critical density as a fundamental organizing principle for ARW dynamics, marking the transition between absorbing inactivity and robustly sustained activity. Its precise value depends on sleep rate, jump bias, dimension, topology, and disordered medium, but is always strictly below full occupancy. The resolution of the density conjecture in one dimension, extension to transitive and more complex graphs, and connection to mixing cutoffs, solidify its role as the universal threshold for sustained activity in ARW and related conservative stochastic systems (Hoffman et al., 3 Jun 2024, Hoffman et al., 29 Jan 2025, Forien, 4 Feb 2025, Junge et al., 12 Sep 2025).