  Numerical computations in strongly-interacting quantum field theories are
often performed using Monte-Carlo sampling methods. A key task in these
calculations is to estimate the value of a given physical quantity from the
distribution of stochastic samples that are generated using the Monte-Carlo
method. Typically, the sample mean and sample variance are used to define the
expectation values and uncertainties of computed quantities. However, the
Monte-Carlo sample distribution contains more information than these basic
properties and it is useful to investigate it more generally. In this work, the
exact form of the probability distributions of two-point correlation functions
at zero momentum in O(N) lattice field theories in the disordered phase and in
infinite volume are determined. These distributions allow for a robust
investigation of the efficacy of the Monte-Carlo sampling procedure and are
shown also to allow for improved estimators of the target physical quantity to
be constructed. The theoretical expectations are shown to agree with numerical
calculations in the O(2) model.
