Monte Carlo estimates of thermal averages and analytic continuation

The Monte Carlo (MC) estimates of thermal averages are usually functions of system control parameters $\lambda $, such as temperature, volume, interaction couplings, etc. Given the MC average at a set of prescribed control parameters $\lambda_{0}$, the problem of analytic continuation of the MC data to $\lambda $-values in the neighborhood of $\lambda_{0}$ is considered in both classic and quantum domains. The key result is the theorem that links the differential properties of thermal averages to the higher-order cumulants. The theorem and analytic continuation formulas expressed via higher-order cumulants are numerically tested on the classical Lennard-Jones cluster system of N=13, 55, and 147 neon particles.