Bifurcation currents for families of group representations in higher rank

Finitely generated groups acting on projective spaces are examples of holomorphic dynamical systems which exhibit a variety of different behaviours. We introduce the notion of proximal stability which measures a form of dynamical stability for the action of a holomorphic family of group representations and we will explain how this property can be detected using a bifurcation current on the parameter space of the family. This bifurcation current measure the pluriharmonicity of the top Lyapunov exponent of the family of representation, defined using a random walk on the group.