  The density of state approach has recently been proposed as a potential route
to circumvent the sign problem in systems at finite density. In this study,
using the Linear Logarithmic Relaxation (LLR) algorithm, we extract the
generalised density of states, which is defined in terms of the imaginary part
of the action, for the self-interacting relativistic lattice Bose gas at finite
density. After discussing the implementation and testing the reliability of our
approach, we focus on the determination of the free energy difference between
the full system and its phase-quenched counterpart. Using a set of lattices
ranging from $4^4$ to $16^4$ , we show that in the low density phase, this
overlap free energy can be reliably extrapolated to the thermodynamic limit.
The numerical precision we obtain with the LLR method allows us to determine
with sufficient accuracy the expectation value of the phase factor, which is
used in the calculation of the overlap free energy, down to values of ${\cal
O}(10^{-480})$. When phase factor measurements are extended to the dense phase,
a change of behaviour of the overlap free energy is clearly visible as the
chemical potential crosses a critical value. Using fits inspired by the
approximate validity of mean-field theory, which is confirmed by our
simulations, we extract the critical chemical potential as the non-analyticity
point in the overlap free energy, obtaining a value that is in agreement with
other determinations. Implications of our findings and potential improvements
of our methodology are also discussed.
