  Inverse probability problems whose generative models are given by strictly
nonlinear Gaussian random fields show the all-or-nothing behavior: There exists
a critical rate at which Bayesian inference exhibits a phase transition. Below
this rate, the optimal Bayesian estimator recovers the data perfectly, and
above it the recovered data becomes uncorrelated. This study uses the replica
method from the theory of spin glasses to show that this critical rate is the
channel capacity. This interesting finding has a particular application to the
problem of secure transmission: A strictly nonlinear Gaussian random field
along with random binning can be used to securely encode a confidential message
in a wiretap channel. Our large-system characterization demonstrates that this
secure coding scheme asymptotically achieves the secrecy capacity of the
Gaussian wiretap channel.
