Bose-Einstein condensation of the dilute hard-sphere Bose gas in the thermodynamic limit, at zero and small positive temperature

Positive temperature (October 5): for every $a>0$ there is $\rho_*(a)>0$ such that for each fixed $0<\rho<\rho_*$ some fixed $T(a,\rho)>0$ gives $\liminf N^{-1}\langle u_{0,L},\gamma^{(1)}_{N,L,T}u_{0,L}\rangle>0$ for the exact canonical hard-sphere Gibbs state as $L\to\infty$, $N/L^3\to\rho$. Ground state (September 24 companion): absolute $\varepsilon_0,c_0>0$ with condensate fraction at least $c_0$ in every ground state when $\rho a^3<\varepsilon_0$. A further companion gives a density-uniform bound for bounded nonnegative finite-range potentials at temperatures up to $\rho^2$. Not shown: the transition temperature, condensation at moderate density, two dimensions, or potentials with attractive parts.

Positive temperature (October 5): for every $a>0$ there is $\rho_*(a)>0$ such that for each fixed $0<\rho<\rho_*$ some fixed $T(a,\rho)>0$ gives $\liminf N^{-1}\langle u_{0,L},\gamma^{(1)}_{N,L,T}u_{0,L}\rangle>0$ for the exact canonical hard-sphere Gibbs state as $L\to\infty$, $N/L^3\to\rho$. Ground state (September 24 companion): absolute $\varepsilon_0,c_0>0$ with condensate fraction at least $c_0$ in every ground state when $\rho a^3<\varepsilon_0$. A further companion gives a density-uniform bound for bounded nonnegative finite-range potentials at temperatures up to $\rho^2$. Not shown: the transition temperature, condensation at moderate density, two dimensions, or potentials with attractive parts.