Burning the plane: densities of the infinite Cartesian grid

Graph burning is a discrete-time process on graphs, where vertices are sequentially burned, and burned vertices cause their neighbours to burn over time. We consider extremal properties of this process in the new setting where the underlying graph is also changing at each time-step. The main focus is on the possible densities of burning vertices when the sequence of underlying graphs are growing grids in the Cartesian plane, centred at the origin. If the grids are of height and width $2cn+1$ at time $n$, then all values in $\left [ \frac{1}{2c^2} , 1 \right ]$ are possible densities for the burned set. For faster growing grids, we show that there is a threshold behaviour: if the size of the grids at time $n$ is $\omega(n^{3/2})$, then the density of burned vertices is always $0$, while if the grid sizes are $\Theta(n^{3/2})$, then positive densities are possible. Some extensions to lattices of arbitrary but fixed dimension are also considered.
Comment: To appear in Graphs and Combinatorics