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arXiv · 2610.27872

Moments of Random Permanents and Hafnians: A Replica Approach

Paolo Facchi·Angelo Mariano·Giorgio Parisi·Saverio Pascazio·Antonello Scardicchio
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Abstract

Using contour-integral representations of the permanent and of the Hafnian, we turn the computation of their moments for random matrices into a replica calculation, analogous to that of the moments of the partition function of a mean-field spin glass. For the $2k$-th moment of $N\times N$ matrices with independent entries, the dominant saddle point is replica symmetric for $k>2$, and we obtain in closed form the exponential growth rate, the prefactor and the first $1/N$ correction, in agreement with exact enumerations. The growth rate depends only on whether the entries are real or complex; higher cumulants affect only the prefactor and the $1/N$ term. At $k=2$ the replica-symmetric solution becomes unstable, separating a power-law bulk of the distribution from a large-deviation tail, which we compute. The results apply to boson sampling with single photons and with squeezed light.

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Paolo Facchi, Angelo Mariano, Giorgio Parisi, Saverio Pascazio, Antonello Scardicchio. 2026-10-20. Moments of Random Permanents and Hafnians: A Replica Approach. https://arxiv.org/abs/2610.27872

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