At the critical probability p_c ≈ 0.5927 for site percolation on the triangular lattice, cluster boundaries become random fractal curves. Oded Schramm (2000) showed these converge to SLE_6 (κ=6), a process defined by the Loewner equation with Brownian driving: dg_t(z)/dt = 2/(g_t(z) − √κ W_t). SLE_κ curves have Hausdorff dimension 1 + κ/8. For κ=6 (percolation), dim = 7/4. Smirnov's 2001 proof (Fields Medal 2010) established conformal invariance rigorously. The connection to 2D turbulence: vorticity isolines follow SLE-like statistics.
Percolation cluster (triangular lattice) — green = open, path = cluster boundary
SLE_κ curve — driven by Brownian motion W_t