 Solution

ID: past-exam-of-the-random-matrix-theory-course-of-the-university-of-oxford/2025/3/b/i/solution

U

replaced by

B (t)

. Pathwise,

e^{i Θ^{+} (t)} = e^{i B (t)}, e^{i Θ^{-} (t)} = e^{- i B (t)}, Θ^{\pm} (t) = \pm B (t) (mod 2 π) .

The real-valued processes

\pm B (t)

give continuous lifts of these angles. The eigenvalues coincide whenever

B (t) \in π Z

; the exponential formulas remain valid at those times.

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