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Random walks
In one dimension, a random walk with t steps of length 1 starting at position 0 can be generated from
NestList[(# + (-1)^Random[Integer])&, 0, t]
or equivalently
FoldList[Plus, 0, Table[(-1)^Random[Integer], {t}]]
A generalization to d dimensions is then
FoldList[Plus, Table[0, {d}], Table[RotateLeft[PadLeft[ {(-1)^Random[Integer]}, d], Random[Integer, d - 1]], {t}]]
A fundamental property of random walks is that after t steps the root mean square displacement from the starting position is proportional to √ t .

The detection distance increases like the square root of the signal strength, covering all 10 11 stars in our galaxy when the signal uses the total power output of a star.

Fairly large terms are sometimes seen quite early: in 5 1/3 term 19 is 3052, while in Root[10 + 8 # - # 3 &, 1] term 34 is 1,501,790.