content_image/357880.jpg content_image/357879.jpg content_image/357878.jpg
appearing in Eq. 1 of the letter, is directly given by the measurement of the third harmonics of the polarisation. 

## II) THIRD HARMONICS FOR A LIQUID WITHOUT GLASSY CORRELATIONS. 

We briefly summarize here what can be expected for the third harmonics in the case of a liquid without any glassy correlations. We use the work of D´ ejardin and Kalmykov [2] which studies the nonlinear dielectric relaxation of an assembly of rigid polar symmetric particles. The particles are assumed to be diluted enough in a non polar solvent, so that any interaction between them can be safely neglected. A strong electric field is applied to the system, which biases the (noninertial) rotational Brownian motion of the polar particles and yields a net dielectric response. This response is expressed by expanding the relaxation functions as a Fourier series in the time domain. Then the infinite hierarchy of recurrence equations for the Fourier components is obtained in terms of a matrix continued fraction. 

We focus on the case where the field is purely ac, namely E ( t ) = E cos ωt . In the following equations, the subscript “D” refers to the calculation of D´ ejardin and Kalmykov. By using their Eq. (21), one gets : 

<img src='content_image/357880.jpg'>

where τ$_{D}$ is the usual Debye time since one finds χ ′′ 1 ,D ∝ ωτ$_{D}$ 1+ ω $^{2}$τ 2 $_{D}$, i.e. χ ′′ 1 is maximum for the frequency f$_{D}$ = 1 / (2 πτ$_{D}$ ). By using Eqs. (7) and (19) of D´ ejardin and Kalmykov, one can calculate all the prefactors and obtain χ$_{3}$$_{,D}$ ( ω = 0): 

<img src='content_image/357879.jpg'>

where N$_{0}$ is the volumic concentration of polar particles, and where, as in the letter, Δ χ$_{1}$ = χ ′ 1 $_{,D}$(0) − χ ′ 1 $_{,D}$( ∞ ). To get a benchmark for the behavior of χ$_{3}$$_{,trivial}$ for the ‘no glassy correlations’ case, we set f$_{D}$ = f$_{α}$ as well as N$_{0}$ = 1 /a 3 with a 3 the molecular volume in glycerol. We thus obtain from the above equations: 

<img src='content_image/357878.jpg'>

