Conformal Invariance & SLE Curves

Schramm-Loewner Evolution traces cluster boundaries at 2D critical points

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Schramm-Loewner Evolution (SLE_κ): the unique conformally invariant random curve family, discovered by Oded Schramm (1999). Parametrized by κ: κ≤4 gives simple curves, 4<κ<8 self-intersecting, κ≥8 space-filling. Critical models map to SLE: percolation hulls → SLE_6 (Smirnov 2001), Ising interfaces → SLE_3, SAW → SLE_{8/3}, uniform spanning tree → SLE_2. The fractal dimension is d_f = 1 + κ/8.