Moments of Random Permanents and Hafnians: A Replica Approach
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× 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.
Related
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BibTeX
@misc{facchi2026moments,
title = {{Moments of Random Permanents and Hafnians: A Replica Approach}},
author = {Paolo Facchi and Angelo Mariano and Giorgio Parisi and Saverio Pascazio and Antonello Scardicchio},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27872},
eprint = {2610.27872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27872},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Moments of Random Permanents and Hafnians: A Replica Approach
AU - Paolo Facchi
AU - Angelo Mariano
AU - Giorgio Parisi
AU - Saverio Pascazio
AU - Antonello Scardicchio
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27872
UR - https://arxiv.org/abs/2610.27872
AB - 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× 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.
ER -