Weak Harris Mixing and Exponential Clustering for Interacting Boundary Laws on Abstract Wiener Spaces

Emmanouil Karolos Čižek (2025). Preprint (submitted / under review).

Abstract

We develop a weak Harris framework for interacting probability measures on abstract Wiener spaces and apply it to exponential clustering (OS4) in Euclidean quantum field theory. Starting from a Gaussian reference measure tilted by a potential on an abstract Wiener space, we formulate quantitative structural hypotheses that encode ellipticity, local C_H^2-regularity, one-sided growth, the existence of a coercive Lyapunov function, and a projected Doeblin minorisation for a discrete-time skeleton of the associated Langevin dynamics. Under these assumptions we prove a weak Harris theorem in an adapted Kantorovich distance W_1^(m), combining control of finitely many low modes with a Lyapunov component, and obtain exponential convergence to equilibrium with constants that are uniform in families of models sharing the same structural data. We then introduce an abstract boundary-law and slab-Markov representation for time-separated observables and show that Harris mixing for the boundary chain implies an OS4 exponential clustering bound for Schwinger function covariances, with a strictly positive decay rate ρ(t) > 0 that is stable under regulator limits.

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