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

By Itai Benjamini, Nicolas Curien. 2012

78 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/636c78a180a356f4737e01ab6a918680128b7661

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Citations in the corpus

Gaussian multiplicative chaos and applications: A review (2013, methodology): “For instance, such a KPZ formula has already been used by physicists [55] to predict the exact values of the Brownian intersection exponents and has been rigorously checked to hold in some special cases (see [26]).”
Random geometry on the sphere (2014, background): “See also Benjamini and Curien [10] for a proof of subdiffusivity of simple random walk on the UIPQ.”
Percolation on uniform infinite planar maps (2013, background): “This spatial Markov property is proved in [5] for uniform triangulations, and a strictly identical proof applies in our setting of quadrangulations.”
Scaling limits for the peeling process on random maps (2014, methodology): “We note that Theorem 1 can be applied to various peeling processes that have been considered in earlier works: peeling along percolation interfaces [3,4], peeling along simple random walk [7], peeling along a Brownian or a SLE6 exploration of the Riemann surface associated with the UIPT [15], etc.”