Seminar: Backbone exponent for two-dimensional percolation

ABSTRACT

We consider Bernoulli percolation in dimension two, obtained by coloring independently, black or white, the vertices of an infinite lattice. Percolation at the critical parameter of phase transition has attracted special interest, because the interfaces of its clusters are conjectured (and proved for the triangular lattice) to be conformally invariant in the scaling limit. In particular, arm events play an instrumental role in describing the geometry of clusters at criticality. Roughly speaking, in any annulus on the lattice, such events require the existence of a given number of crossing paths, whose colors are specified by some sequence. The asymptotic behavior of “polychromatic” arm events, when arms of both colors are present in the sequence, is now very well understood. We discuss the monochromatic case, i.e. the case when all arms have the same color.

More precisely, we derive an exact expression for the celebrated backbone exponent, corresponding to two disjoint black arms. This exponent was first considered in statistical physics in the 1970’s, and determining its value had remained an open question since then. It turns out to be a root of an elementary function, and contrary to previously-known arm exponents for 2D percolation, which are all rational, it has a transcendental value. More specifically, we use techniques which have been developed recently to compute the conformal radii of random domains defined by SLE curves, based on the coupling between SLE and Liouville quantum gravity (LQG), and using crucially input from Liouville conformal field theory (LCFT).

This talk is based on a joint work with Pierre Nolin, Xin Sun and Zijie Zhuang.

BIOGRAPHY

Wei Qian is assistant professor in the department of Mathematics and the department of Physics of City University Hong Kong. She is currently on leave from the position of Chargée de recherche in CNRS in France. Prior to that, Wei Qian obtained her PhD at ETH Zürich, and was subsequently a Junior Research Fellow at Churchill college in the University of Cambridge. Wei Qian’s works focus on two dimensional random geometry, in relation to mathematical physics.