Forest Fire Model on $\mathbb{Z}_{+}$ with Delays

TL;DR

This paper extends the classic forest-fire model on $\mathbb{Z}_+$ by allowing nonzero burning times $\theta_{x,i}$ and random spreading delays $\Delta_{x,i}$, and studies how these delays affect the emergence of an infinite fire. It shows that when spreading delays are strictly positive with nonzero probability, an infinite fire occurs almost surely; in the instant-burning regime ($\theta=0$) with constant or suitably distributed $\Delta$, the infinite-fire index can be bounded or characterized, and a sharp threshold in the asymptotic mean delay $\mathbb{E}[\Delta_x]$ determines the presence of infinite fires. Conversely, when the spread is instantaneous ($\Delta=0$) but burning times are positive, no infinite fire occurs and the first-passage times grow logarithmically in the burnt region, with precise tail bounds and several conjectures on second-burn times and extreme-growth behavior. The results illuminate how delays and burning durations govern subcritical versus supercritical propagation in delayed forest-fire processes and connect to coupled percolation-type dynamics and known stochastic-fire models.

Abstract

We consider a generalization of the forest fire model on $\mathbb{Z}_+$ with ignition at zero only, studied in [arXiv:0907.1821]. Unlike that model, we allow delays in the spread of the fires as well as the non-zero burning time of individual ``trees''. We obtain some general properties for this model, which cover, among others, the phenomena of an ``infinite fire'', not present in the original model.

Figures (3)

• Figure 1: Graphical representation in the case $\theta\equiv 0$, and a constant $\Delta\equiv a>0$. Solid arrows represent the spread of fire, while dotted arrows indicate unsuccessful attempts to spread. The black dots mark the appearance of trees, and green lines denote periods during which a tree occupies a site before it burns. The highlighted line of arrows represents the infinite fire.
• Figure 2: The black dots correspond to the appearance of trees; the green (red resp.) segments are the periods when a site is occupied by a "healthy" (burning resp.) tree. The shaded areas represent the periods when a burning tree at site $x$ affects site $x+1$. The solid arrows show the fire spreading to a neighbouring tree; the dotted arrows represent the situations when the fire tried to spread unsuccessfully.
• Figure 3: how the second fire reaching $m_i$ can spread very far.

Theorems & Definitions (29)

• Remark 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['t:infirexists']}
• Theorem 3.1
• Remark 3.2
• proof
• Theorem 3.3
• proof