Lyapunov exponents and stability properties of higher rank representations

Generalizing results of Deroin-Dujardin, we introduce the notion of proximal stability for a holomorphic family $(\rho_{\lambda})$ of representations $\Gamma \to \text{SL}(d, \mathbb{C})$, where $\Gamma$ is a finitely generated group, and show that it is equivalent to the pluriharmonicity of Lyapunov exponents of the family (defined using random walks).