The Dynamic Range of LZ

The electronics of the LZ experiment, the 7-tonne dark matter detector to be installed at the Sanford Underground Research Facility (SURF), is designed to permit studies of physics where the energies deposited range from 1 keV of nuclear-recoil energy up to 3,000 keV of electron-recoil energy. The system is designed to provide a 70% efficiency for events that produce three photoelectrons in the photomultiplier tubes (PMTs). This corresponds approximately to the lowest energy threshold achievable in multi-tonne time-projection chambers, and drives the noise specifications for the front end. The upper limit of the LZ dynamic range is defined to accommodate the electroluminescence (S2) signals. The low-energy channels of the LZ amplifiers provide the dynamic range required for the tritium and krypton calibrations. The high-energy channels provide the dynamic range required to measure the activated Xe lines

The outliers of a deformed Wigner matrix

We derive the joint asymptotic distribution of the outlier eigenvalues of an additively deformed Wigner matrix $H$. Our only assumptions on the deformation are that its rank be fixed and its norm bounded. Our results extend those of [The isotropic semicircle law and deformation of Wigner matrices. Preprint] by admitting overlapping outliers and by computing the joint distribution of all outliers. In particular, we give a complete description of the failure of universality first observed in [Ann. Probab. 37 (2009) 1-47; Ann. Inst. Henri Poincar\'{e} Probab. Stat. 48 (1013) 107-133; Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices. Preprint]. We also show that, under suitable conditions, outliers may be strongly correlated even if they are far from each other. Our proof relies on the isotropic local semicircle law established in [The isotropic semicircle law and deformation of Wigner matrices. Preprint]. The main technical achievement of the current paper is the joint asymptotics of an arbitrary finite family of random variables of the form $\langle\mathbf{v},(H-z)^{-1}\mathbf{w}\rangle$.Comment: Published in at http://dx.doi.org/10.1214/13-AOP855 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Eigenvector Distribution of Wigner Matrices

We consider $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $\nu_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution $\nu_{ij}$ coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector-eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk

A Lower Bound on the Ground State Energy of Dilute Bose Gas

Consider an N-Boson system interacting via a two-body repulsive short-range potential $V$ in a three dimensional box $\Lambda$ of side length $L$. We take the limit $N, L \to \infty$ while keeping the density $\rho = N / L^3$ fixed and small. We prove a new lower bound for its ground state energy per particle $$\frac{E(N, \Lambda)}{N} \geq 4 \pi a \rho [ 1 - O(\rho^{1/3} |\log \rho|^3) ],$$ as $\rho \to 0$, where $a$ is the scattering length of $V$.Comment: 26 pages, AMS LaTe