Generalized exchange cluster algorithm to compute efficiently covariances and susceptibilities in Monte Carlo

We present a Monte Carlo method to compute efficiently susceptibilites or covariances of two physical variables. The method relies on a generalization of the exchange cluster algorithm to any model of interacting particles with any $2$-body interactions. The principle is to select clusters of variables belonging to two independent replicas of the system. An improved estimator of the covariance of two physical variables (in one replica) is then proposed. This estimator has the zero-variance property in the limit wh ere these variables are independent.In practice the scaling of the statistical fluctuations as a function of the number of degrees of freedom $N$ is reduced from $O(N^2)$ to $O(N)$. This lower scaling is illustrated on a Lennard Jones model.