The determinant and the permanent are defined by the same formula — a sum over permutations of products of matrix entries — except the permanent uses no signs. This tiny difference makes the determinant computable in polynomial time and the permanent #P-hard. One of the sharpest complexity separations in mathematics.

Hunter, Kwan & Sauermann (arXiv:2603.15856) show that over finite fields, the permanent of a random matrix is significantly more uniform in its distribution than the determinant. The quantity that’s harder to compute is easier to predict statistically. The determinant, despite being tractable, has a less uniform distribution.

This is counterintuitive because computational difficulty usually correlates with statistical unpredictability. Hard-to-compute functions tend to behave like pseudorandom generators precisely because their output can’t be anticipated. But the permanent breaks this association: its output is well-distributed even though computing any individual value is intractable. The computational barrier isn’t about the output’s statistical structure — it’s about the path from input to output.

The distinction matters because it reveals that complexity and uniformity are independent axes. A function can be simultaneously hard to evaluate and easy to characterize in aggregate. The difficulty isn’t in what the permanent does but in how you’d verify what it did for any specific matrix.