Exponential bounds of the condensation for dilute Bose gases

We consider N bosons on the unit torus $\mathrm{\Lambda }=[0,1{]}^3$ in the Gross-Pitaevski regime where the interaction potential scales as $N^{2}V(N(x-y))$. We prove that the thermal equilibrium at low temperatures exhibits the Bose-Einstein condensation in a strong sense, namely the probability of having $n$ particles outside of the condensation decays exponentially in $n$.