Persistence of spectral projections for stochastic operators on large tensor products

TL;DR

This paper develops a perturbation theory for spectral projections of stochastic operators on large tensor products, showing their smooth persistence under parameter changes, with applications to cellular automata dynamics.

Contribution

It introduces a rigorous framework for the smooth persistence of spectral projections in stochastic operators depending on parameters, extending perturbation theory to this context.

Findings

Spectral projections depend smoothly on parameters for stochastic operators.

A perturbation theory for operators with spectral gap is established.

Application to effective slow dynamics in cellular automata.

Abstract

In this paper it is proved that for families of stochastic operators on a countable tensor product, depending smoothly on parameters, any spectral projection persists smoothly, where smoothness is defined using norms based on ideas of Dobrushin. A rigorous perturbation theory for families of stochastic operators with spectral gap is thereby created. It is illustrated by deriving an effective slow 2-state dynamics for a 3-state probabilistic cellular automaton. Some further potential applications are discussed.