Frog Model with Death and Drift

• Frog model with death and drift is an interacting particle system on graphs where active frogs perform biased random walks and may die, influencing recurrence behavior.
• Analytical techniques, including recursive distributional equations and coupling with branching processes, establish rigorous recurrence and transience criteria.
• The model informs phase transition analysis in networks and has applications in studying epidemic spread, rumor transmission, and other spatial propagation phenomena.

The frog model with death and drift is a class of interacting particle systems on graphs or lattices that incorporates both random walk bias (drift) and the possibility of particle death. The model variants have been studied extensively in recent years, particularly on trees and lattices, with rigorous conditions for recurrence, transience, and phase transitions under varying drift and death parameters. The introduction of drift counteracts natural recurrence in low dimensions, while death truncates particle trajectories; the interplay of these mechanisms governs whether the origin or root is visited infinitely often by activated particles.

1. Formal Definition and Dynamics

Consider a graph $G$ (commonly $\mathbb{Z}^d$ or a rooted $d$-ary tree $\mathbb{T}_d$), with one active frog initially at a distinguished vertex (the “origin” in $\mathbb{Z}^d$ or “root” in trees) and a configuration of sleeping frogs placed at other vertices (typically deterministically or via an i.i.d. distribution $\mu$). Activated frogs perform nearest-neighbor random walks according to a transition kernel $p(\cdot)$, with a nonzero mean drift:

$m = \sum_e p(e) e = a e_1, \qquad a \in (0, 1)$

for the drift toward the $e_1$ direction. In tree models, the bias $p$ is the probability to move toward the root and $1-p$ to a uniformly chosen child. Death is introduced via a survival probability $q < 1$; before each jump, a frog is killed with probability $1-q$.

Once an active frog lands on a vertex holding sleeping frogs, all sleeping frogs are activated and proceed independently by the same rules.

2. Recurrence and Transience Criteria

Recurrence refers to the property that the origin (or root) is visited infinitely often by activated frogs. In the drift-only case on $\mathbb{Z}^d$, the main criterion is (Döbler et al., 2014):

$$E_\mu[ (\log^+(\eta_x))^{(d+1)/2} ] = \infty \implies \text{recurrence}$$

where $\log^+(y) = \max\{ \log(y), 0 \}$, and $\mu$ is the common law of sleeping frogs. For $d=1$, this reduces to $E_\mu[\log^+(\eta_1)] = \infty$, a sharp threshold matching earlier results.

The drift typically suppresses recurrence by pushing typical trajectories away from the origin. However, sufficiently heavy-tailed initial configurations compensate: even though individual frogs have subpolynomial probabilities of returning (scaling as $1/(y_1-x_1)^{(d-1)/2}$), an explosion in the activated frog population enables many aggregate returns.

With death, the process becomes more delicate. For example, on biregular trees $\mathbb{T}_{d_1, d_2}$, a phase transition at a critical value $p_c$ satisfies

$$p_c(\mathbb{T}_{d_1, d_2}) = \frac{1}{2} + \Theta\Big( \frac{1}{d_1} + \frac{1}{d_2} \Big) \quad \text{as } d_1, d_2 \to \infty$$

(Lebensztayn et al., 2018). For $p < p_c$, extinction occurs almost surely; for $p > p_c$, survival (and recurrence) has positive probability.

3. Influence of Drift and Death on the Model

Drift modifies the transition kernel such that frogs tend to move preferentially in a direction (e.g., toward or away from the origin/root). Strong drift increases transience, requiring heavier tails in the initial sleeping distribution or increased “upward” bias for recurrence.

Death truncates frog paths, reducing the effective population propagating through the graph. The recurrence-transience threshold is not monotone in the drift parameter when death is present; an optimal, intermediate drift may maximize recurrence probability, contrary to previous monotonicity conjectures (Ahmed et al., 21 Oct 2025). For example, with $d$-ary trees and death parameter $q \in [0.9999998, 1]$, recurrence is possible only for a narrow drift window $p \in [0.4950, 0.4975]$.

4. Mathematical Formulations and Analytical Techniques

The analysis relies on stochastic orders (e.g., increasing-concave and probability generating function orders), recursive distributional equations, coupling to auxiliary particle systems (non-backtracking or self-similar frog models), and percolation or branching process theory.

On trees, recurrence thresholds are determined using recursive frog models (RFM) and second-moment arguments (Paley–Zygmund inequality); the expectation recursion typically takes the form

$$E[V_t] = \rho q^3 (1 + P(A_t^y | A_t^x)) E[V_{t-1}] + \rho q^2$$

with thinning due to both drift ($\rho = p/(1-p)$) and death ($q$). Moment estimates and stochastic bounds on random walk excursion lengths (dominated by geometric random variables with explicit decay parameters in $p$) ensure convergence properties for critical thresholds.

The death-modified random walk is given by

$P(S_{n+1} = y | S_n = x) = q \cdot p_0(x, y)$

with $q$ survival probability and $p_0(x, y)$ transition probability. Many recurrence results are established by comparing the process to a branching random walk or a percolation process.

5. Phase Transition and Monotonicity Phenomena

The frog model with death and drift displays phase transitions separating regions of almost sure extinction from survival/recurrence. On biregular trees, explicit lower and upper bounds for $p_c$ are derived analytically via embedded Galton–Watson branching processes, percolation theory, and computation of hitting probabilities.

Recent results emphasize that the recurrence probability is not monotone in the drift: excessive bias can suppress returns by preventing sleeping frog activation along branches, while insufficient drift allows death to dominate. Only intermediate drift magnitudes yield positive recurrence probability (Ahmed et al., 21 Oct 2025).

6. Implications, Applications, and Future Directions

The interplay between drift and death in frog models informs the paper of spatial spreading phenomena in percolation, epidemic models, rumor transmission in networks, or combustion. The delicate balance controlling phase transitions—regimes where drift and death compensate or exacerbate each other—provides insight for designing and controlling propagation in networked systems.

Techniques developed for these models (recursive thinning, stochastic orders, coupling to auxiliary processes) are robust and adaptable for more general interacting random walks with site or particle disorder, spatial heterogeneity, or additional features such as random lifetimes (Carvalho et al., 12 Mar 2025).

Advances in bounding critical drift and survival thresholds, monotonicity studies, and recursive coupling methods lay the groundwork for further research into interacting particle systems, the geometry of spread, and the identification or control of phase transitions in more complex environments.

7. Figures and Parameter Regime Diagrams

Recent studies present explicit phase diagrams for the frog model with death and drift on trees, showing recurrence only in exceedingly narrow windows of drift and death parameters. For example, in (Ahmed et al., 21 Oct 2025), Figure 1 displays a (p, q) diagram for the binary tree ($d=2$), shading transient regions and indicating the narrow strip ($p \in [0.4950, 0.4975],\ q \in [0.9999998, 1]$) where recurrence is established.

Such diagrams illustrate the nontrivial and non-monotonic relationship between drift, death, and recurrence, emphasizing that in models with both mechanisms, optimal recurrence requires careful tuning of both parameters.

---

The frog model with death and drift represents a rich domain for rigorous probabilistic analysis, phase transition identification, and the paper of branching–activation propagation processes in structured and random environments. The balance between drift, initial configuration, and particle death governs long-term global behavior, with recurrence criteria tightly linked to heavy-tailed distributions, drift geometry, and recursive activation mechanisms.