Divergent four-point dynamic density correlation function of a glassy suspension

We use a diagrammatic formulation of the dynamics of interacting Brownian particles\footnote{G. Szamel, J. Chem. Phys. \textbf{127}, 084515 (2007).} to study a four-point dynamic density correlation function of a glassy colloidal suspension. We re-sum a class of diagrams which separate into two disconnected components upon cutting a single propagator. The resulting formula for the four-point correlation function can be expressed in terms of three-point functions closely related to the three-point susceptibility introduced by Biroli \textit{et al.}\footnote{G. Biroli \textit{et al.}, Phys. Rev. Lett. \textbf{97}, 195701 (2006).} and the standard two-point correlation function. We numerically evaluate the four-point function and the associated dynamic correlation length. Both the amplitude of the four-point function and the correlation length diverge at the mode-coupling transition.