Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points

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 (Geom Funct Anal 13:935-974, 2003) to the quadrangulation case. Our main result is that, up to polylogarithmic factors, n³ 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 Knizhnik et al. (Modern Phys Lett A 3:819-826, 1988) 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.