  We study the pioneer points of the simple random walk on the uniform infinite
planar quadrangulation (UIPQ) using an adaptation of the peeling procedure of
Angel to the quadrangulation case. Our main result is that, up to
polylogarithmic factors, $n^3$ pioneer points have been discovered before the
walk exits the ball of radius $n$ in the UIPQ. As a result we verify the KPZ
relation in the particular case of the pioneer exponent and prove that the walk
is subdiffusive with exponent less than 1/3. Along the way, new geometric
controls on the UIPQ are established.

Probability