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?