The Bayes Principle and Segal Axioms for $P(\phi)_2$, with application to Periodic Covers

Understand

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.