Suppose one has a link diagram of the unknot, and applies random Reidemeister moves
until the unknot is reached.
Surely it requires an exponential number of moves, exponential in, say, the crossing number
of the original diagram?
The 2001 Hass-Lagarias paper, "The number of Reidemeister moves needed for unknotting,"
established an exponential upper bound on the number of moves needed, but I am not finding
a result on the expected number of *random* moves needed.
I would like affirmation that not only is it hard, but one would not easily stumble into
a solution, because then it would not truly be hard!
(This in the spirit of Gower's much more substantive MO question,
"Are there any very hard unknots?")

A reference would be appreciated! Thanks!

*Edit*: Apologies for the flawed question (thanks to Ryan Budney for clarifying it).
I had in mind the expected number of
random moves to reach the unknot from a random (in some sense!) diagram of the unknot.

**Answered.** The question has been answered in the comments by Theo Johnson-Freyd and
Ori Gurel-Gurevich: the expected number of moves is $\infty$! As Ori put it,

for any starting diagram of the unknot, there is a positive probability of never unknotting it.

doestake an exponential number of moves in general. Moreover, I would guess that if you only make random moves, it's highly unlikely you will ever find a route to the unknot -- it seems more likely that you would be endlessly lost in diagram space. – Ryan Budney Oct 9 '11 at 1:12