$T \to 0$ mean-field population dynamics approach for the random 3-satisfiability problem

During the past decade, phase-transition phenomena in the random 3-satisfiability (3-SAT) problem has been intensively studied by statistical physics methods. In this work, we study the random 3-SAT problem by the mean-field first-step replica-symmetry-broken cavity theory at the limit of temperature $T\to 0$. The reweighting parameter $y$ of the cavity theory is allowed to approach infinity together with the inverse temperature $\beta$ with fixed ratio $r=y / \beta$. Focusing on the the system's space of satisfiable configurations, we carry out extensive population dynamics simulations using the technique of importance sampling and we obtain the entropy density $s(r)$ and complexity $\Sigma(r)$ of zero-energy clusters at different $r$ values. We demonstrate that the population dynamics may reach different fixed points with different types of initial conditions. By knowing the trends of $s(r)$ and $\Sigma(r)$ with $r$, we can judge whether a certain type of initial condition is appropriate at a given $r$ value. This work complements and confirms the results of several other very recent theoretical studies.