In this note, we study the asymptotics of an integral on the unitary group first proposed by Mergny and Potters as a multiplicative counterpart to the well-known Harish-Chandra Itzykson Zuber integral. In particular we prove in a mathematically rigorous manner a result from Mergnys and Potterss original paper in the case $beta =1,2$ and we generalize it for multiple arguments.
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