Moments of Random Permanents and Hafnians: A Replica Approach

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 $1\mathrm{/}N$ 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 $1\mathrm{/}N$ 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.