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Why is lattice physics on a torus?
It depends on what you are looking at. For example, correlation functions in the bulk (by this I basically mean "far away from the boundary") do not really care about boundary conditions, or whether there are boundaries at all. The correction can be quantified. In the "worst" scenario, if the system is gapless, boundary effect introduces corrections to the correlation functions that asymptotically decay as a power-law with the distance to the boundary. Therefore, as long as the boundary is kept far enough (i.e. further than any length scales you care about), the error is negligible. If the system is gapped, then the correction is even smaller, decaying as an exponential function of the distance to the boundary. Of course, things can change a lot when you approach the boundary. For example, in some exotic materials (usually called "topological" xxx, i.e. topological insulators), often one finds low-energy modes that only live on boundary. But even in these cases, if you measure local properties in the bulk it is still not affected much by the boundary.
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Alter PDF: All You Need to Know
D. [1] 2] 3] Theoretical Physics, 3rd edition, by D. D. M. S.