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We show that the Ginzburg-Weinstein diffeomorphism $\mathfrak{u}(n)^* \toU(n)^*$ of Alekseev-Meinrenken admits a scaling tropical limit on an open densesubset of $\mathfrak{u}(n)^*$. The target of the limit map is a product$\mathcal{C} \times T$, where $\mathcal{C}$ is the interior of a cone, $T$ is atorus, and $\mathcal{C} \times T$ carries an integrable system with naturalaction-angle coordinates. The pull-back of these coordinates to$\mathfrak{u}(n)^*$ recovers the Gelfand-Zeitlin integrable system ofGuillemin-Sternberg. As a by-product of our proof, we show that the Lagrangiantori of the Flaschka-Ratiu integrable system on the set of upper triangularmatrices meet the set of totally positive matrices for sufficiently largeaction coordinates.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

The $U(n)$ Gelfand-Zeitlin system as a tropical limit of Ginzburg-Weinstein diffeomorphisms