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Questioning the relationship between the $χ$4 susceptibility and the dynamical correlation length in a glass former

Clusters of fast and slow correlated particles, identified as dynamical heterogeneities (DHs), con-stitute a central aspect of glassy dynamics. A key ingredient of the glass transition scenario is asignificant increase of the cluster size $ξ$4 as the transition is approached. In need of easy-to-computetools to measure $ξ$4 , the dynamical susceptibility $χ$4 was introduced recently, and used in various ex-perimental works to probe DHs. Here, we investigate DHs in dense microgel suspensions using imagecorrelation analysis, and compute both $χ$4 and the four-point correlation function G4 . The spatialdecrease of G4 provides a direct access to $ξ$4 , which is found to grow significantly with increasingvolume fraction. However, this increase is not captured by $χ$4 . We show that the assumptions thatvalidate the connection between $χ$4 and $ξ$4 are not fulfilled in our experiments.