 0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1 0  0.5  1  1.5  2  2.5 r max  ($^{λ}$ ) -0.8 -0.6 -0.4 -0.2  0  2  2.5  3  3.5  4 log 10  r max  ($^{λ}$ c ) log$_{10}$ N 7 On the light of these results, one may wonder how r$_{max}$ ( λ$_{c}$ ) behaves in the thermody- namical limit. As it is very difficult to do exact calculations beyond size N > 10000, we will rely on the finite size scaling analysis to extrapolate its behavior for N → ∞ . In Fig. 4 we show how this quantity evolves with the size N of the environment. We plot results for the a small region around λ ≈ λ$_{c}$ where it becomes sharp and size dependent. Therefore, the decoherence factor behaves in a critical way around λ = λ$_{c}$ where r$_{max}$ ( λ$_{c}$ ) undergoes a dip towards zero which is sharper and deeper for larger values of N . FIG. 4: r $_{max}$( λ $_{c}$) in function of the size of the environment N , in a double logarithmic scale. Squares represent the case α = 0; circles, α = 0 . 4; upper triangles, α = 0 . 6; lower triangles, α = 0 . 7. The straight lines correspond to a least squares fit to the power law r $_{max}$( λ $_{c}$) = AN − 1 / $^{4}$. FIG. 3: (Color online). r $_{max}$( λ ) i n function of the coupling λ , for different values of α and N . Black lines represent the case α = 0; dark grey (red online) lines, α = 0 . 4; grey (magenta online), α = 0 . 6; and light grey (cyan onlin e), α = 0 . 7. Solid lines represent N = 10000; dotted lines, N = 2500; and dashed lines, N = 6 00.