  We point out a precise connection between Brownian motion, Chern-Simons
theory on S^3, and 2d Yang-Mills theory on the cylinder. The probability of
reunion for N vicious walkers on a line gives the partition function of
Chern-Simons theory on S^3 with gauge group U(N). The probability of starting
with an equal-spacing condition and ending up with a generic configuration of
movers gives the expectation value of the unknot. The probability with
arbitrary initial and final states corresponds to the expectation value of the
Hopf link. We find that the matrix model calculation of the partition function
is nothing but the additivity law of probabilities. We establish a
correspondence between quantities in Brownian motion and the modular S- and
T-matrices of the WZW model at finite k and N. Brownian motion probabilitites
in the affine chamber of a Lie group are shown to be related to the partition
function of 2d Yang-Mills on the cylinder. Finally, the random-turns model of
discrete random walks is related to Wilson's plaquette model of 2d QCD, and the
latter provides an exact two-dimensional analog of the melting crystal corner.
Brownian motion provides a useful unifying framework for understanding various
low-dimensional gauge theories.
