Q ≈ 0, while those that stay within this distance (“immobile”) have Q ≈ 1. As a function of strain γ , the average over- lap 〈 Q 〉 will decay, when particle displacements u$_{na}$ are comparable to the probing length-scale a . The overlap function is similar to the intermedi- ate scattering function with wave-vector q ∼ 1 /a . Thus, a sets the probing length-scale. The decay of Q ( γ, a ) then gives an associated structural re- laxation strain, γ $^{⋆}$( a ), on which particle positions decorrelate. In the following we are interested in the dynami- cal heterogeneity of Q and the fluctuations around its average value The strength of the correlations are encoded in the peak-height, h ( γ ) ≡ χ$_{4}$ ( a $^{⋆}$( γ ) , γ ) (black circles in Fig. 7)). As χ$_{4}$ can be written as the integral over a correlation function, it is connected to the correlation volume, or to the number of correlated particles. Assuming that this volume forms a com- pact region in space 45,46 we can relate the am- plitude of χ$_{4}$ to a dynamic correlation length via ξ $^{2}$( γ ) = h ( γ ). The second surprising feature in Fig. 8 is the system-size dependence of h . It turns out that h/N rather than h itself is independent of system- size, indicating a finite variance of the distribu- tion of Q values in the thermodynamic limit (see Eq. (4)). Assuming the connection with the corre- lation length to hold, ξ 2 ∼ h , this implies a cor- relation length that is proportional to the length of the simulation box, ξ ≈ 0 . 3 L , independent of volume-fraction and distance to point J. This illus- trates the fact that quasistatic dynamics is inher- First, by rescaling the strain-axis with the aver- age length of elastic branches, Δ γ$_{avg}$ (see Fig. 6) we find reasonable scaling collapse for all studied volume-fractions and system-sizes. This implies that cooperativity, as measured by the amplitude of χ$_{4}$ and the length of elastic branches are intimately related. The frequency of plastic events sets the strain-scale for dynamical heterogeneities. As the length of the elastic branches decreases with system size, the absolute maximum shifts towards smaller strains, with h ( γ ) becoming effectively a decreasing function of γ in the thermodynamic limit. There are two surprising features in this plot. Following the maxima in Fig. 7 from left to right, one sees that the amplitude first increases and then quickly drops to small values. This implies that there is a finite strain γ , at which χ$_{4}$ presents an ab- solute maximum. To extract this maximum we plot in Fig. 8 the amplitude h ( γ ) for various volume- fractions φ and system-sizes N . Figure 8: Amplitude h ( γ ) as taken from the qua- sistatic simulations. Data for different volume- fractions and system-sizes. The axes are nor- malized according to the scaling form h ( γ ) = N ˜ h ( γ/ Δ γ$_{avg}$ ), with Δ γ$_{avg}$ taken from Fig 6. which defines the (self-part of the) dynamical sus- ceptibility χ$_{4}$ . This is displayed in Fig. 7 as a func- tion of both strain γ and probing length-scale a . For each γ it has a well defined peak (at a $^{⋆}$( γ )) that parallels the decay of the overlap function 〈 Q 〉 . 31 χ$_{4}$ ( a, γ ) = N ( 〈 Q ( γ, a ) $^{2}$〉 − 〈 Q$_{a}$ ( γ, a ) 〉 2 ) , (4) Figure 7: Dynamical susceptibility χ$_{4}$ as function of probing length-scale a for various different strains γ = (5 , 20 , 50 , 200 , 500 , 2000) · 10 $_{−}$5 (from left to right) and φ = 0 . 9. The maxima of the curves (black circles) define the amplitude h ( γ ). 8  0  50  100  150  200 10 -4 10 -3 10 -2 10 -1 a  0.02  0.04  0.06  0.08 10 -1 10 0 10 1 10 2 h/N avg