Time-delay matrix, midgap spectral peak, and thermopower of an Andreev   billiard
  We derive the statistics of the time-delay matrix (energy derivative of the scattering matrix) in an ensemble of superconducting quantum dots with chaotic scattering (Andreev billiards), coupled ballistically to $M$ conducting modes (electron-hole modes in a normal metal or Majorana edge modes in a superconductor). As a first application we calculate the density of states $\rho_0$ at the Fermi level. The ensemble average $\langle\rho_0\rangle=\delta_0^{-1}M[\max(0,M+2\alpha/\beta)]^{-1}$ deviates from the bulk value $1/\delta_0$ by an amount depending on the Altland-Zirnbauer symmetry indices $\alpha,\beta$. The divergent average for $M=1,2$ in symmetry class D ($\alpha=-1$, $\beta=1$) originates from the mid-gap spectral peak of a closed quantum dot, but now no longer depends on the presence or absence of a Majorana zero-mode. As a second application we calculate the probability distribution of the thermopower, contrasting the difference for paired and unpaired Majorana edge modes. 
M. Marciani, P. W. Brouwer and C. W. J. Beenakker
cond-mat.mes-hall