Fig. 6.

Higher-order fluctuations reveal the transition to QLE. Left panel: an analogous version of the neutral collapse plot in Fig. 4 for the higher-order LD moment ${\sigma }_d^4(f_0)$. Symbols denote the results of forward time simulations for a range of recombination rates, which are colored by the corresponding value of f₀. The solid black line shows the prediction from the perturbation expansion in Equations (D22) and (D23), and the dashed lines indicate the position, $NR{f}_0\sim 1/f_0$, where the perturbation expansion is predicted to break down. The solid colored lines show the asymptotically matched predictions from Equation (F5), which capture the transition to the quasi-linkage equilibrium regime. Right panel: the conditional distribution of the double-mutant frequency for fixed values of the marginal mutation frequencies, $f_A\approx f_B\approx f_0$. Colored lines show forward-time simulations for pairs of neutral mutations, in which the marginal frequencies of both mutations were observed in the range $0.13\le f_A,f_B\le 0.17$; the double-mutant frequency was further downsampled to n = 200 individuals to enhance visualization. The dashed lines indicate the approximate positions of linkage equilibrium ($f_{AB}\approx f_0^2$; left) and perfect linkage ($f_{AB}\approx f_0$; right). Conditional distributions are shown for three different recombination rates, whose characteristic shapes illustrate the transition between the mutation-dominated ($NR{f}_0\ll 1$; orange), clonal recombinant ($1\ll NR{f}_0\ll 1/f_0$; green) and QLE ($NR{f}_0\gg 1/f_0$; blue) regimes.