A variance formula related to a quantum conductance problem

Let t   be a block of an Haar-invariant orthogonal (β=1β=1), unitary (β=2β=2) or symplectic (β=4β=4) matrix from the classical compact groups O(n)O(n), U(n)U(n) or Sp(n)Sp(n), respectively. We obtain a close form for Var(tr(t∗t))Var(tr(t∗t)). The case for β=2β=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 (8β)−1(8β)−1 for β=1,2β=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.