Hole probability for non-interacting fermions in a d-dimensional trap

The hole probability, i.e., the probability that a region is void of particles, is a benchmark of correlations in many body systems. We compute analytically this probability P (R) for a sphere of radius R in the case of N noninteracting fermions in their ground state in a d-dimensional trapping potential. Using a connection to the Laguerre-Wishart ensembles of random matrices, we show that, for large N and in the bulk of the Fermi gas, P (R) is described by a universal scaling function of kF R, for which we obtain an exact formula (kF being the local Fermi wave-vector). It exhibits a super exponential tail P (R) / e-κd(kF R)d+1 where κdis a universal amplitude, in good agreement with existing numerical simulations. When R is of the order of the radius of the Fermi gas, the hole probability is described by a large deviation form which is not universal and which we compute exactly for the harmonic potential. Similar results also hold in momentum space.

The Buffon needle problem for Lévy distributed spacings and renewal theory

What is the probability that a needle dropped at random on a set of points scattered on a line segment does not fall on any of them? We compute the exact scaling expression of this hole probability when the spacings between the points are independent identically distributed random variables with a power-law distribution of index less than unity, implying that the average spacing diverges. The theoretical framework for such a setting is renewal theory, to which the present study brings a new contribution. The question posed here is also related to the study of some correlation functions of simple models of statistical physics.