Electronic Physics Auxiliary Publication Service: Supplementary Information for : Caroline Crauste-Thibierge, Coralie Brun, Francois Ladieu, Denis L’Hˆ ote, Giulio Biroli, Jean-Philippe Bouchaud Evidence of growing spatial correlations at the glass transition from nonlinear response experiments For the sake of completness, we detail in the four sections hereafter: (i) the connexion between the third harmonics of the fundamental response and the nonlinear part of the response; (ii) what can be expected for the third harmonics response of a dielectric liquid where no (glassy) correlations are present; (iii) some additionnal information about our experimental setup; (iv) how the upper limit of the heating contribution was computed (Fig. 4 of the letter) and the reduction which can be expected from a more refined estimate. I) NONLINEAR RESPONSE MEASUREMENTS THROUGH THIRD HARMON- ICS DETECTION As in the main paper, we consider the nonlinear response of a dielectric system to a time dependent electric field E ( t ). The most general relationship between the polarisation P ( t ) and the excitation E ( t ) is a series expansion in E [1]. For a purely ac field, the even terms are forbidden because of the E ( t ) → − E ( t ) symmetry, which yields : In this equation ϵ$_{0}$ is the dielectric constant of vacuum, χ$_{1}$ the linear susceptibility and χ$_{3}$ the cubic nonlinear susceptibility in the time domain. The dots in Eq. 3 indicate an infinite sum involving higher order nonlinear susceptibilities χ$_{5}$ , etc. Note that causality implies 12 P ( t ) ϵ$_{0}$ = ∫ ∞ −∞ χ$_{1}$ ( t − t $^{′}$) E ( t $^{′}$) dt $^{′}$+ ∫ ∞ −∞ χ$_{3}$ ( t − t ′ $_{1}$, t − t ′ $_{2}$, t − t ′ $_{3}$) × E ( t ′ $_{1}$) E ( t ′ $_{2}$) E ( t ′ $_{3}$) dt ′ $_{1}$dt ′ $_{2}$dt ′ $_{3}$+ .... (3)