Local Gaussian domination and Bose condensation in lattice boson model

We use infrared bounds method to study the existence of a Bose condensation phase transition in a three-dimensional lattice model of interacting bosons. Upper bound (local Gaussian domination) is presented on the Bogolyubov inner product of creation and annihilation bosonic operators in momentum space. We focus on the situation with a non-negative chemical potential. The presence of a Bose condensation is established for sufficiently small superstable interaction potential. A special case of the model considered is the Bose-Hubbard model.