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

Abstract

We consider the Dirichlet eigenvalues of the Laplacian among a Poissonian cloud of hard spherical obstacles of fixed radius in large boxes of R d , d ≥ 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 . We show here that with probability arbitrarily close to 1 as ℓ goes to infinity, the spectral gap stays bigger than σ(log ℓ) −(1+2 d) , where the small positive number σ depends on how close to 1 one wishes the probability. Incidentally, the scale (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.

Mini-Workshop: A Geometric Fairytale full of Spectral Gaps and Random Fruit

“…Lastly, the Kac-Luttinger conjecture in the sense of a type-I g-BEC occurrence in probability has now also been confirmed in the case of hard Poissonian obstacles in dimensions d ≥ 2, in the very recent work [17]. We would also like to mention that in any dimensions the physical intuition favors the occurrence of type-I g-BEC in probability in the case of soft Poissonian obstacles as well [11,12].…”