DLR Equations for the Superstable Bose Gas at any Temperature and Activity

We construct a thermodynamic limit for the grand canonical Bose gas in dimension $d⩾1$ (in its Feynman-Kac representation) with superstable interaction at any inverse temperature $\beta >0$ and any chemical potential $\mu \in \mathbb{R}$. Our infinite volume model is naturally a distribution over configurations of finite loops and possibly interlacements. We prove the limiting process to solve a new class of DLR equations involving random permutations and Brownian paths.