General form of the equilibrium distribution under exQLE

Determine the general analytic form of the stationary equilibrium distribution Q(χ) for the extended quasi-linkage equilibrium (exQLE) cumulant dynamics at an arbitrary truncation order K by solving the forward Kolmogorov equation for cumulants and imposing the stationarity condition j(χ) = 0, and characterize how this equilibrium depends on cumulants, mutation rates, and recombination rates.

Background

The paper introduces an extended quasi-linkage equilibrium (exQLE) framework that allows cumulants up to order K to evolve dynamically while assuming higher-order cumulants rapidly equilibrate. This yields deterministic and stochastic equations of motion for cumulants driven by the gradient of average fitness through a symmetric positive (semi)definite matrix D(χ).

At finite population size, the cumulant dynamics induce a Fokker–Planck equation over cumulant space. The authors note that, analogous to prior analyses under standard QLE, one can, in principle, derive an explicit equilibrium distribution Q(χ) from the forward Kolmogorov equation by setting the probability current j(χ) to zero. They provide partial insight: for K = 2, the equilibrium includes interactions between first-order cumulants in an exponential form, and, similar to QLE, features an exponential form with an entropic term.

However, the authors explicitly state that the general form of the equilibrium distribution for exQLE remains conjectural. Resolving this would clarify the stationary behavior of cumulants under selection, mutation, and recombination in the exQLE regime and provide a foundation for further analytical and inferential developments.