This generates a spanning web. A spanning web is like a spanning tree, except that it allows closed loops that encircle the root. For a finite system, it's possible that there are no encircling loops, which makes the system identical to a spanning tree. However, for an infinite system, there are encircling loops with probability one. Spanning webs are explained in this paper. (In the notation of that paper, we have y=2.)
Spanning webs are interesting because they can be used to understand the behavior of a kinetically constrained model of dimers on a square lattice.
Click on the "New Web" button to create a new spanning web with the specified x and y dimensions, and x and y position of the root. (The root is the special site that closed loops are allowed to encircle.) Note that if you put the root on the boundary, no encircling loops are possible, and so the spanning web will necessarily be a spanning tree.
The analysis relating spanning webs to the kinetically constrained model of dimers on a square lattice makes us particularly interested in random walks, starting at the root of the spanning web. Numerical simulations show that such random walkers undergo anomalous diffusion. However, we only know the anomalous diffusion exponent numerically. (Even the anomalous diffusion exponent for the simpler case of random walks on spanning trees is not analytically known.)
The random walker always starts at the root. A trail of black sites shows the trail from the current position of the random walker to its initial position, while the green sites show all sites that the random walker has visited.
You can choose to show either all bonds, or just the tree connected to the root, and the encircling loop(s). All bonds "drain into" either the root or an encircling loop(s). More than one encircling loop is possible, but for the small system sizes that we can reasonably draw on a computer screen, it's rare to get two or more loops.
The six fields at the bottom right show: