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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, 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 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.

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