  The peeling process is an algorithmic procedure that discovers a random
planar map step by step. In generic cases such as the UIPT or the UIPQ, it is
known [Curien & Le Gall, Scaling limits for the peeling process on random maps,
Ann. Inst. Henri Poincar\'e Probab. Stat. 53, 1 (2017), 322-357] that any
peeling process will eventually discover the whole map. In this paper we study
the probability that the origin is not swallowed by the peeling process until
time $n$ and show it decays at least as $n^{-2c/3}$ where \[c \approx
0.12831235141783245423674486573872854933142662048339843...\] is defined via an
integral equation derived using the Lamperti representation of the spectrally
negative $3/2$-stable L\'evy process conditioned to remain positive [Chaumont,
Kyprianou & Pardo, Some explicit identities associated with positive
self-similar Markov processes, Stochastic Process. Appl. 119, 3 (2009),
980-1000] which appears as a scaling limit for the perimeter process. As an
application we sharpen the upper bound of the sub-diffusivity exponent for
random walk of [Benjamini & Curien, Simple random walk on the uniform infinite
planar quadrangulation: subdiffusivity via pioneer points, Geom. Funct. Anal.
23, 2 (2013), 501-531].
