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

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