Stepping paths II

Bartholomew

Randomly fill a very large grid of squares white or black so that each square has a 50% chance of being white or black. You can approximately do this in Paint using the airbrush tool.

The fill tool can turn one "sector" of white squares black, or one "sector" of black squares white. (zoom in 8x to see this clearly; you can also turn on "show grid" from the view-->zoom--> menu)

What is the average size of one of these sectors?

Of those sectors that enclose other sectors entirely within them, what is the average number of sectors the latter sectors enclose?

On average, how many other sectors is a given sector in contact with?

Is there a branch of math that deals with this?