Cloning Algorithms: from Large Deviations to Population Dynamics

Population dynamics provides a numerical tool allowing for the study of rare events by means of simulating a large number of copies of the system, supplemented with a selection rule that favours the rare trajectories of interest. The cloning algorithm allows the estimation of a large deviation function (LDF) of additive observables in Markov processes. However, such algorithms are plagued by finite simulation time $t$ and finite population size $N_c$ effects that can render their use delicate. First, using a non-constant population approach, we analyze the small-$N_c$ effects in the initial transient regime. These effects play an important role in the numerical determination of LDF. We show how to overcome these effects by introducing a time delay in the evolution of populations, additional to the discarding of the initial regime of the population growth where these discreteness effects are strong. Then, the study of the finite-$t$ and finite-$N_c$ scalings in the LDF evaluation is done using two different versions of the algorithm, in discrete and continuous-time. We show that these scalings behave as $1/N_c$ and $1/t$ in the large-$N_c$ and large-$t$ asymptotics respectively. Moreover, we show that one can make use of this convergence speed in order to extract the asymptotic behavior in the infinite-$t$ and infinite-$N_c$ limits resulting in a better LDF estimation. These scalings are later generalized and evidence of a breakdown for large-size systems is presented.