Pierre Nolin, Wei Qian, Xin Sun, Zijie Zhuang: Backbone exponent for two-dimensional percolation


We derive an exact expression for the celebrated backbone exponent for 2D Bernoulli percolation at criticality. It turns out to be a root of an elementary function, and contrary to previously-known arm exponents for this model, which are all rational, it has a transcendental value. Our derivation relies on the connection to the SLE bubble measure, the coupling between SLE and Liouville quantum gravity (LQG), and the integrability of Liouville conformal field theory (LCFT). Along the way, we derive a formula not only for \kappa=6 (corresponding to percolation), but in fact for all \kappa \in (4,8).

More specifically, we use techniques which have been developed to compute the conformal radii of random domains defined by SLE curves, based on the coupling between SLE and LQG. Compared to prior methods, a crucial new input is the exact solvability of structure constants in LCFT. Various quantities related to percolation, which can be expressed in terms of conformal radii, can thus be computed in this way.



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