01 · The idea
In everyday language
Color sites of a triangular lattice randomly with equal probabilities, and ask whether one color crosses a given region. At the critical value, clusters appear at every scale. Physicists predicted that as the lattice spacing shrinks, crossing probabilities and interfaces should respect conformal maps, transformations preserving local angles. Smirnov found special discrete observables that satisfy analogues of complex-analytic relations. Their continuum limits made conformal invariance rigorous for critical triangular-lattice percolation and, through related work, the planar Ising model. This explains how a discrete model with only a few lattice symmetries can develop a much richer symmetry in its large-scale limit.
Why it matters
These proofs established rigorous foundations for key predictions of two-dimensional statistical physics. They connect lattice models to complex analysis and stochastic Loewner evolution, allowing exact formulas and universal-looking behavior to be understood mathematically.
