I'm a little confused with how a single random variable E in [0,1] determines a bounce. In PathTracer, our sampling functions use two random variables to determine the wi direction, and some wi's (such as for mirror) are uniquely determined. So shouldn't something like a point in hypercube [0,1]^(2k) give us a path? Does this not depend on how we are sampling directions for the bounce? Or is this only for the 2D case?
Based on the side, I think the example is only for the 2D case (note that each bounce is determined by mapping E to theta, and no phi is selected). I agree that in the 3D case (as we're sampling for PathTracer) it should be [0,1]^(2k).