  We construct a $P(\phi)_2$ Gibbs state on infinite volume periodic surfaces
(namely, with discrete ``time translations'') by analogy with 1-dimensional
spin chains and establish the mass gap for our Gibbs state, there are no phase
transitions. We also derive asymptotic properties of the $P(\phi)_2$ partition
function on certain towers of cyclic covers of large degrees that converge to
the periodic surface in some appropriate sense. This gives the first
construction of an interacting Quantum Field Theory on surfaces of infinite
genus with a mass gap. The main ingredient in our approach is to reconcile the
so-called $P(\phi)_2$ model from classical constructive quantum field theory
(CQFT) with Riemannian version of the axioms proposed by G. Segal in the 90's.
We show the $P(\phi)_2$ model satisfies these axioms, appropriately adjusted.
One key ingredient in our proof is to use what we call ``the Bayes principle''
of conditional probabilities in the infinite dimensional setting. We also give
a precise statement and full proof of the locality of the $P(\phi)_2$
interaction.
