A variance formula related to a quantum conductance problem

Abstract

Let t be a block of an Haar-invariant orthogonal ( β = 1 ), unitary ( β = 2 ) or symplectic ( β = 4 ) matrix from the classical compact groups O ( n ) , U ( n ) or Sp ( n ) , respectively. We obtain a close form for Var ( tr ( t ∗ t ) ) . The case for β = 2 is related to a quantum conductance problem, and our formula recovers a result obtained by several authors. Moreover, our result shows that the variance has a limit ( 8 β ) −1 for β = 1 , 2 and 4 as the sizes of t go to infinity in a special way. Although t in our formulation comes from a block of an Haar-invariant matrix from the classical compact groups, the above limit is consistent with a formula by Beenakker, where t is a block of a circular ensemble.