We will prove the conjecture of Tsirelson, that 2D critical percolation is a noise in the scaling limit. Thus we construct the first example of a black noise in the plane.
The result is a joint work with Oded Schramm and has two parts:
First we propose a new approach to constructing the scaling limit of percolation in the plane: a random collection of rectangles which are crossed. We show that for many percolation models there is tightness and hence subsequential scaling limits, and that for critical site percolation on triangular lattice the limit is unique.
The second part proves that (in the scaling limit) when a domain is cut by a smooth curve, percolation configuration in it can be reconstructed from configurations in two halves (i.e. there is no information "stored" on the curve). On the lattice it means that resampling the configuration in a small neighborhood of the curve and slightly perturbing it off the curve does not change the crossing properties much.
Integrable open spin chains related to the quantum affine algebras U_q(o(3)) and U_q(A_2^(2)) are considered. Hamiltonians (and other integrals) belong to the Birman-Wenzl-Murakami algebra (BMW). The symmetry algebra U_(o(3)) and the BMW-algebra centralize each other in the spin chain representation space. Consequently, the multiplet structure of the energy spectra is obtained.