Fig. 5
Histogram of Δi,t(α,Γ,Ω) for small cities (N < 10000) for α = 0.5 (left panels) and α = 0.75 (rigth panels). Moreover, we consider different intervals of Γ: [0.0001,0.005] (upper panels), [0.005,0.05] (central panels),[0.05,0.09] (lower panels). According to the statistical-mechanics theory, the data in the left panels are fitted with a Gaussian distribution \(y = e^{ - x^2/(2\sigma ^2)}/\sqrt {2\pi \sigma ^2}\) while those in the right panels with a quartic exponential distribution \(y = e^{ - x^4/\sigma ^4}/[2\sigma \Gamma (5/4)]\). The Gaussian fits (blue lines in the left panels) with one free parameter σ give a decreasing coefficient of determination R2 from e to a showing that approaching Γ = 0 a critical point is expected. In particular, the parameters of the fits are: σ = 0.10 ± 0.03 R2 = 0.956 (a), σ = 0.12 ± 0.01 R2 = 0.976 (c), σ = 0.10 ± 0.01 R2 = 0.981 (e). The quartic exponential fits (red lines in the right panels) with one free parameter σ provide instead an increasing R2 from f to b, confirming the theoretical scenario that requires that the correct normalization at the critical point has exponent 3/4. The parameters of the fits are: σ = 0.15 ± 0.01, R2 = 0.999 (b), σ = 0.078 ± 0.007, R2 = 0.991 (d), σ = 0.08 ± 0.01, R2 = 0.586 (e)
