No, all of the child's memory mappings are shared with its parent. Upon write, though, that becomes a new committed mapping; I believe the entire page is copied, so you could end up committing another 4k with a one-byte write from the child.

Hm. Then suppose I turn off the OOM killer. What happens to a process that writes to its parent's page if the system has no memory left to commit?

COW is usually a kind of loophole. Most systems only count the pages once (otherwise you'd have to reject the fork, but that becomes inconvenient fast).