Ginibre Matrix Permanent Density Proves Anticoncentration Conjecture

Frederic Koehler and Pui Kuen Leung have proven that the normalized row-ordered permanent of a Ginibre matrix possesses a radial density satisfying a specific equation, resolving a longstanding mathematical problem. Ginibre matrices, introduced in a 1965 paper, are defined as matrices with independent, standard Gaussian entries, scalars with independent real and imaginary parts, distinguishing them from more commonly studied Hermitian random matrices. This work directly addresses the Permanent Anticoncentration Conjecture, initially proposed by Aaronson and Arkhipov in connection with BosonSampling [AA13]. The researchers demonstrate this result by comparing the squared Gaussian permanent with the squared determinant in Laplace-transform order, offering a concrete advancement in the understanding of random matrix theory and its connections to quantum simulation. A decades-old…