Interpretation of Vacuum diagram

Question

When I read QFT textbook, I had some puzzle abou the vacuum to vacuum diagram

$$\langle 0| \exp (-i\int^{+T}_{-T} dtH_{I}) | 0 \rangle= \exp(\sum {\rm vacuum ~diagrams}) $$

I wonder how to represent the vacuum diagrams like 8 in terms of particle path? Does time interval $2T$ appear?

How about partition function in the Euclidean space? consider 2$D$ Euclidean space,

$$ ds^2=\rho^2 d\tau^2+d\rho^2 $$

what is the Feymann diagram of the partition function? or the particle path in the above 2$D$ space?

Comments

the question is not very clear for me ... Is it about the frequency discretisation to get a sum instead of an integral in the computation of the partition function ?

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It is about giving the lines in the diagram a temporal interpretation. In such an interpretation, closed lines would seem to correspond to a compact time integral whereas in the above formula $T\to\infty$. But in reality, the temporal interpretation of Feynamn diagrams is just for picturing things and cannot be given a strict interpretation. Thus a proper answer to the question would consist in explaining how a figure 8 diagram in $\Phi^4$ theory defines a corresponding term in the bare perturbation series.