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

TL;DR

This paper investigates the spectral gap in a Poissonian obstacle model and demonstrates a form of Bose-Einstein condensation in a related bosonic system, establishing probabilistic bounds on spectral properties and phase transition behavior.

Contribution

It provides new probabilistic bounds on the spectral gap in the Kac-Luttinger model and links these results to Bose-Einstein condensation phenomena in disordered systems.

Findings

Spectral gap stays larger than a specific logarithmic scale with high probability.

The scale $( ext{log } l)^{-(1+ 2/d)}$ likely captures the true size of the spectral gap.

A type-I generalized Bose-Einstein condensation occurs in the model for high densities.

Abstract

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 $(\log l)^{-2/d}$. We show here that with probability arbitrarily close to $1$ as $l$ goes to infinity, the spectral gap stays bigger than $\sigma (\log l)^{-(1 + 2/d)}$, where the small positive number $\sigma$ 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.