A Variance Formula Related to a Quantum Conductance Problem

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 V ar(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 (8β)−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.