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For any edge weight distribution, we consider the uniform spanning tree (UST)on finite graphs with i.i.d. random edge weights. We show that, for boundeddegree expander graphs and finite boxes of ${\mathbb Z}^d$, the diameter of theUST is of order $n^{1/2+o(1)}$ with high probability, where $n$ is the numberof vertices.

Diameter of uniform spanning trees on random weighted graphs