  We study a one parameter family of discrete Loewner evolutions driven by a
random walk on the real line. We show that it converges to the stochastic
Loewner evolution (SLE) under rescaling. We show that the discrete Loewner
evolution satisfies Markovian-type and symmetry properties analogous to SLE,
and establish a phase transition property for the discrete Loewner evolution
when the parameter equals 4.
