Random minimal surfaces in spheres

For any dimension $n>1$, we construct a branched minimal immersion ψₙ from a closed Riemann surface Σₙ to the round n-sphere of radius √8, such that if Σₙ is endowed with the pullback metric and if K is its Gaussian curvature, then Σₙ is almost hyperbolic in the sense that limn→ ∞ 1/Area(Σₙ)∫Σₙ |K+1|=0 and Σₙ Benjamini-Schramm converges to the hyperbolic plane. Our proof is based on a connection between minimal surface theory and random matrix theory. The maps ψₙ are obtained by applying the spherical Plateau problem to random unitary representations ρN of the free group F₂.