Random planar curves, conformal maps, universality classes: κ=2 (loop-erased RW), 3 (Ising), 4 (DGFF), 6 (percolation), 8 (UST)
Loewner equation: ∂g_t/∂t = 2/(g_t(z)−W_t), W_t = √κ · B_t (Brownian driver)
d_f = 1 + κ/8 (fractal dim); phase: κ≤4 simple curve, 4<κ<8 self-intersecting, κ≥8 space-filling
SLE_κ (Schramm 2000): the unique family of conformally invariant random curves in the upper half-plane. The Loewner chain g_t maps ℍ\γ[0,t] → ℍ conformally, driven by Brownian motion scaled by √κ. Key universality: κ=2 loop-erased RW (Lawler-Schramm-Werner), κ=3 Ising interfaces, κ=4 free Gaussian field level lines, κ=6 critical percolation (Smirnov's theorem, Fields Medal 2010), κ=8 uniform spanning tree. For κ≤4: simple curve. κ>4: curve hits itself. κ≥8: space-filling.