  Gaussian Boson Sampling is a popular method for experimental demonstrations
of quantum advantage, but many subtleties remain in fully understanding its
theoretical underpinnings. An important component in the theoretical arguments
for approximate average-case hardness of sampling is anticoncentration, which
is a second-moment property of the output probabilities. In Gaussian Boson
Sampling these are given by hafnians of generalized circular orthogonal
ensemble matrices. In a companion work [arXiv:2312.08433], we develop a
graph-theoretic method to study these moments and use it to identify a
transition in anticoncentration. In this work, we find a recursive expression
for the second moment using these graph-theoretic techniques. While we have not
been able to solve this recursion by hand, we are able to solve it numerically
exactly, which we do up to Fock sector $2n = 80$. We further derive new
analytical results about the second moment. These results allow us to pinpoint
the transition in anticoncentration and furthermore yield the expected linear
cross-entropy benchmarking score for an ideal (error-free) device.
