I am trying to have a better understanding of the population annealing algorithm. In the section POPULATION ANNEALING, the steps of the algorithms are described. In Step 2 (Resampling (Split/Remove) of the replicas:), the new replica $x^k_i$ is resampled according to the probability $p^k$. So, what does it actually mean? Can I use a biased coin toss with the probability $p^k$ for heads and remove the replica if it is a tail?
The paper is rather explicit. It states that the weights $W^k$ are used to calculate the probability to resample a particular replica, $P^k = W^k /(\sum_k W^k)$. In particular it says, "Thus, a replica with a small weight $W^k$ is removed with a high probability, while a replica with a large weight tends to have multiple ''descendants''." Hence, your picture is correct, if you are interested only in the case whether or not a particular state resampled, because then a Binomial distribution is applicable. However, in general a better picture is that you have a urn with many balls and each ball has a number printed on it. The weights tell you how many balla have a particular number -- this is the multinomial distribution.