Random minimal surfaces in spheres

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