On the spectral gap in the Kac–Luttinger model and Bose–Einstein condensation

We consider the Dirichlet eigenvalues of Laplacian among a Poissonian cloud hard spherical obstacles fixed radius in large boxes $\mathbb{R}^d$, $d \ge 2$. In box side-length $2l$ centered at origin, lowest eigenvalue is known to be typically order $(\log l)^{-2/d}$. show here that with probability arbitrarily close $1$ as $l$ goes infinity, spectral gap stays bigger than $\sigma (\log l)^{-(1 + 2/d)}$, where small positive number $\sigma$ depends on how one wishes probability. Incidentally, scale l)^{-(1+ 2/d)}$ expected capture correct size gap. Our result involves proof new deconcentration estimates. Combining this lower bound results Kerner-Pechmann-Spitzer, we infer type-I generalized Bose-Einstein condensation for Kac-Luttinger system non-interacting bosons impurities, sole macroscopic occupation one-particle ground state when density exceeds critical value.