  Characterizing the scaling with the total particle number ($N$) of the
largest eigenvalue of the one--body density matrix ($\lambda_0$), provides
informations on the occurrence of the off-diagonal long-range order (ODLRO)
according to the Penrose-Onsager criterion. Setting $\lambda_0\sim
N^{\mathcal{C}_0}$, then $\mathcal{C}_0=1$ corresponds to ODLRO. The
intermediate case, $0<\mathcal{C}_0<1$, corresponds for translational invariant
systems to the power-law decaying of (non-connected) correlation functions and
it can be seen as identifying quasi-long-range order. The goal of the present
paper is to characterize the ODLRO properties encoded in $\mathcal{C}_0$ [and
in the corresponding quantities $\mathcal{C}_{k \neq 0}$ for excited natural
orbitals] exhibited by homogeneous interacting bosonic systems at finite
temperature for different dimensions. We show that $\mathcal{C}_{k \neq 0}=0$
in the thermodynamic limit. In $1D$ it is $\mathcal{C}_0=0$ for non-vanishing
temperature, while in $3D$ $\mathcal{C}_0=1$ ($\mathcal{C}_0=0$) for
temperatures smaller (larger) than the Bose-Einstein critical temperature. We
then focus our attention to $D=2$, studying the $XY$ and the Villain models,
and the weakly interacting Bose gas. The universal value of $\mathcal{C}_0$
near the Berezinskii--Kosterlitz--Thouless temperature $T_{BKT}$ is $7/8$. The
dependence of $\mathcal{C}_0$ on temperatures between $T=0$ (at which
$\mathcal{C}_0=1$) and $T_{BKT}$ is studied in the different models. An
estimate for the (non-perturbative) parameter $\xi$ entering the equation of
state of the $2D$ Bose gases, is obtained using low temperature expansions and
compared with the Monte Carlo result. We finally discuss a double jump
behaviour for $\mathcal{C}_0$, and correspondingly of the anomalous dimension
$\eta$, right below $T_{BKT}$ in the limit of vanishing interactions.
