Fig. 4: Experimental and numerical results.
a, Experiments. The logarithm of relaxation time τ is plotted versus the logarithm of the correlation length ξ in natural swarms (logarithms are in base 10). The critical exponent zexp is the slope of the linear fit. Because experimental uncertainty affects both τ and ξ, standard LS regression (which assumes no uncertainty on the abscissa) systematically underestimates the exponent. RMA regression treats uncertainty on the two variables in a symmetric way by minimizing the sum of the areas of the triangles formed by each point and the fitted line. The inset shows, for a sample point, the construction of the triangle whose area is given by |ΔxΔy|/2 (see Methods). RMA regression gives zexp = 1.37. The physical ranges for τ and ξ are 80 ms < τ < 610 ms and 50 mm < ξ < 250 mm (Table I of Supplementary Section VI). b, Numerical simulations. Plot of logτ versus \({\log \xi}\) in the ISM. Numerical errors are so small that LS and RMA give the same result, zsim = 1.35, and the LS error is ±0.04. c, Experimental resampling. To estimate the error bar on the experimental exponent we use a resampling method. We randomly draw 107 subsets with half the number of points and in each subset we determine z using RMA. We report here three such random subsets (orange, selected point; grey, unselected point). Rare experimental fluctuations under resampling can produce an unphysical value of z smaller than 1; this, however, happens in only 0.002% of the data subsets. d, Final comparison. Probability distribution of the experimental critical exponent (orange) from the resampling method of c. The standard deviation of this distribution gives the error on the experimental exponent, ±0.11. The vertical band (purple) indicates the position and error of the numerical critical exponent, and coloured symbols indicate the various RG fixed-point values of z.
