We study site percolation on a square lattice with random compact diamond-shaped neighborhoods. Each site $s$ is connected to others within a neighborhood in the shape of a diamond of radius $r_s$, where $r_s$ is uniformly chosen from the set $\{i,i+1,...,m\}$ with $i\le m$. The model is analyzed for all values of $i=0,...,7$ and $m=i,...,10$, where $\overline{z}(i,m)$ denotes the average number of neighbors per site and $p_c(i,m)$ is the critical percolation threshold. For each fixed $i$, the product $\overline{z}(i,m)p_c(i,m)$ is found to converge to a constant as $m\to \infty$. Such behavior is expected when $i=m$ (single diamond sizes), for which the product $z\left(i\right)p_c\left(i\right)$ tends toward $2^d{\eta }_c$, where ${\eta }_c$ is the continuum percolation threshold for diamond-shaped regions or aligned squares in two dimensions ($d=2$). This case is further examined for $i=1,...,10$, and the expected convergence is confirmed. The particular case $i=m$ was first studied numerically by Gouker and Family in 1983. We also study the relation to systems of deposited diamond-shaped objects on a square lattice. For monodisperse diamonds of radius $r$, there is a direct mapping to percolation with a diamond-shaped neighborhood of radius $2r+1$, but when there is a distribution of object sizes, there is no such mapping. We study the case of the deposition of mixtures of diamonds of radius $r=0$ and $r=1$, and contrast it with the $(i,m)=(1,2)$ model and also the continuum percolation of disks of two sizes.

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• Percolation