This can be recast as a random walk on a line. Let $n_t$ be the number of amoebae after $t$ events, and process the events in any order which makes sense. (It may help to think of this as serialising a parallel process on a single-core CPU). For example, you could choose to number the the amoebae by the $t$ at which they are created and always process the lowest-numbered living amoeba. Now at each event $t$ we can decrement the number of amoebae, increment it, or do nothing.
To see informally that a random walk on an infinite line with equiprobable steps of $\{-1, 0, 1\}$ eventually hits zero, suppose that at event $t$ it hasn't yet hit zero, i.e. $n_t > 0$. Then by symmetry it's 50/50 whether it will reach zero or $2n_t$ first. Avoiding eventually hitting zero is like tossing a fair coin infinitely many times without ever getting tails.
Peter Taylor
·
2021-08-05T07:50:35Z (over 3 years ago)