Shift of the condensation temperature caused by weak repulsion

For the dilute 3D Bose gas, prove that $(T_{c} - T_{c}^{0})/T_{c}^{0} = c\rho^{1/3}a + o(\rho^{1/3}a)$, with $T_{c}^{0}$ the ideal-gas value, and determine c (lattice Monte Carlo gives c approximately 1.3, Arnold and Moore; Kashurnikov, Prokof'ev and Svistunov, 2001). Answer: the existence of the phase transition and the value of c.

A proof of a phase transition at $T_{c}$ with rigorous bounds on $c$.