  There is a strong interest in studying the correspondence between Euclidean
quantum fields and neural networks. This correspondence takes different forms
depending on the type of networks considered. In this work, we study this
correspondence in the case of deep Boltzmann machines (DBMs) having a tree-like
topology. We use p-adic numbers to encode this type of topology. A p-adic
continuous DBM is a statistical field theory (SFT) defined by an energy
functional on the space of square-integrable functions defined on a p-adic
N-dimensional ball. The energy functionals are non-local, meaning they depend
on the interaction of all the neurons forming the network. Each energy
functional defines a probability measure. A natural discretization process
attaches to each probability measure a finite-dimensional Boltzmann
distribution, which describes a hierarchical DBM. We provide a mathematically
rigorous perturbative method for computing the correlation functions. A
relevant novelty is that the general correlation functions cannot be calculated
using the Wick-Isserlis theorem. We give a recursive formula for computing the
correlation functions of an arbitrary number of points using certain
3-partitions of the sets of indices attached to the points.
