  We study the effect of Markov moves on $L^2$-Burau maps of braids, in order
to construct link invariants from these maps with a process mirroring the
well-known Alexander-Burau formula.
  We prove such a Markov invariance for $L^2$-Burau maps going to the groups of
the braid closures or deeper. In the case of the link group, the corresponding
link invariant was known to be the $L^2$-Alexander torsion by a previous result
of A. Conway and the author.
  Furthermore, we find two counter-examples, meaning two families of
$L^2$-Burau maps that cannot yield link invariants in this way. The proofs use
relations between Fuglede-Kadison determinants, Mahler measures, and random
walks on Cayley graphs, and the works of Boyd, Bartholdi and Dasbach-Lalin.
  All of this suggests that twisted $L^2$-Alexander torsions are the only link
invariants we can hope to construct from $L^2$-Burau maps with such a process.
