  Signal detection is one of the main challenges of data science. As it often
happens in data analysis, the signal in the data may be corrupted by noise.
There is a wide range of techniques aimed at extracting the relevant degrees of
freedom from data. However, some problems remain difficult. It is notably the
case of signal detection in almost continuous spectra when the signal-to-noise
ratio is small enough. This paper follows a recent bibliographic line which
tackles this issue with field-theoretical methods. Previous analysis focused on
equilibrium Boltzmann distributions for some effective field representing the
degrees of freedom of data. It was possible to establish a relation between
signal detection and $\mathbb{Z}_2$-symmetry breaking. In this paper, we
consider a stochastic field framework inspiring by the so-called "Model A", and
show that the ability to reach or not an equilibrium state is correlated with
the shape of the dataset. In particular, studying the renormalization group of
the model, we show that the weak ergodicity prescription is always broken for
signals small enough, when the data distribution is close to the
Marchenko-Pastur (MP) law. This, in particular, enables the definition of a
detection threshold in the regime where the signal-to-noise ratio is small
enough.
