Effect of vertex weights on spatial-to-mean-field transition

Determine how vertex weights affect the transition between spatial and mean-field behavior in finite-range or spread-out finite-torus ILRFPP as the interaction range \(\ell_n\) grows with the torus size.

Background

For finite-range torus models, the interaction range $\ell_n$ may grow with the system size, producing a possible transition between spatial and mean-field asymptotics. The paper does not determine how the heavy-tailed vertex environment modifies this transition.

This question applies both to the finite-range cost and to truncated power-law costs on the torus.