|
Résumé |
We construct the continuum analogue of the chemical distance metric in
lattice models such as percolation. The chemical distance metric is the
graph distance induced by the percolation clusters. It is known that for
critical percolation, the lengths have non-trivial scaling behaviour,
however it is very difficult to find the exact scaling exponent. (This
is one of the questions from Schramm's ICM 2006 article that remains
unsolved.)
In a joint work with Valeria Ambrosio and Jason Miller, we construct a
chemical distance metric on the CLE gasket for each $\kappa\in]4,8[$. We
show that it is unique metric that is geodesic, Markovian, and
conformally covariant. The characterisation is reminiscent of the LQG
metric, but our objects behave very differently, and hence our
techniques also differ significantly from those used in LQG. For
$\kappa=6$, we conjecture that our random metric space is the scaling
limit of critical percolation. |
