I produced the following image using a statistical simulation and am wondering if the resulting shapes have an existing name:
(Blue indicates where the field is near zero, green where it is positive, and red where it is negative)
These shapes strike me as similar to spherical harmonics, but they are decidedly not that.
If it helps, the image specifically represents the difference between the expected final position of a random walk where a single wall is placed (in between the bright areas of the second and third lobes) and the same random walk without the wall. The random walk consists of 8,000 steps on a discrete grid. (This results in checkering where it is impossible to visit every other grid cell with an even number of steps.) Each step can be in one of the +x, +y, -x, or -y directions. At the wall, either the +y or -y direction is removed from the pool of candidates depending on which side of the wall the original particle is at. Technically the image is not zero anywhere, just very close to zero ($< 0.001$ times the peak) at the blue spots.
I have experimented with random equations in this Desmos document to try and find some level curves with mixed success. Nothing I've tried matches perfectly.
