Dynamical coupling between Ising and FK percolation

TL;DR

This paper explores the construction of Markov processes that couple Ising spin dynamics with FK percolation configurations, revealing limitations and proposing both local and non-local solutions.

Contribution

It introduces new local and non-local dynamics for coupling Ising and FK percolation, highlighting feasibility constraints and offering a contracted localized version.

Findings

Local dynamics fulfilling only spin marginal condition

Non-local dynamics satisfying spin and coupling conditions

Feasibility constraints for coupled dynamics

Abstract

We investigate the problem of constructing a dynamics on edge--spin configurations which realizes a coupling between a Glauber dynamics of the Ising model and a dynamical evolution of the percolation configurations. We dream of constructing a Markov process on edge--spin configurations which is reversible with respect to the Ising--FK coupling measure, and such that the marginal on the spins is a Glauber dynamics, while the marginal on the edges is a Markovian evolution. We present two local dynamics, one which fulfills only the first condition and one which fulfills the first two conditions. We show next that our dream process is not feasible in general. We present a third dynamics, which is non local and fulfills the first and the third conditions. We finally present a localized version of this third dynamics, which can be seen as a contraction of the first dynamics.