Characteristic length scale of the inhomogeneous mode-coupling theory: beyond scaling predictions

The inhomogenous mode-coupling theory of Biroli \textit{et al.}\ [Phys. Rev. Lett. \textbf{97}, 195701 (2006)] allows for the identification of a characteristic length scale that diverges as the mode-coupling transition is approached. We numerically investigate this length scale as a function of time, wave-vector, and distance from the transition by examining the small $\mathbf{q}$ expansion of the dynamic susceptibility $\xi_{\mathbf{q}}(\mathbf{k};t)$ defined by Biroli \textit{et al.} We confirm the scaling predictions of Biroli \textit{et al.}. In addition, we show that the characteristic length is in qualitative agreement with simulations where the length scale is obtained from four-point correlation functions. Finally, we show that the length scale has virtually no $k$ dependence and thus it is well defined. The $k$-independence of the length contrasts with the very strong $k$ dependence of $q\to 0$ limit of the dynamic susceptibility.