Random harmonic maps into spheres

Understand

Let $S$ be a punctured Riemann surface with Euler characteristic $\chi(S)<0$.

• For any unitary representation $\rho: \pi_1(S) \to U(N)$, we introduce its renormalized energy and its harmonic representatives, which are equivariant harmonic maps from the universal cover of $S$ to the unit sphere in $\mathbb{C}^N$.
• Our main result is that if a sequence of unitary representations $\rho_j$ strongly converges, then their renormalized energies converge to $\frac{\pi}{4}|\chi(S)|$ and the shape of their harmonic representatives converges to a unique rescaled hyperbolic metric.
• Combining this statement with examples of strongly converging representations provided by random matrix theory, we derive the following applications.