# One-sided versus two-sided stochastic descriptions

## TL;DR

This paper explores the relationship between one-sided and two-sided stochastic models, showing that under weak dependence conditions, the equivalence between Markov chains and Markov fields no longer holds, with a counterexample based on long-range Ising models.

## Contribution

It demonstrates that the equivalence between Markov chains and Markov fields breaks down under weak dependence, providing a counterexample involving long-range Ising models.

## Key findings

- Equivalence holds under strong dependence but breaks down with weak dependence.
- Counterexample based on entropic repulsion in long-range Ising models.
- Neither class contains the other under relaxed dependence conditions.

## Abstract

It is well-known that discrete-time finite-state Markov Chains, which are described by one-sided conditional probabilities which describe a dependence on the past as only dependent on the present, can also be described as one-dimensional Markov Fields, that is, nearest-neighbour Gibbs measures for finite-spin models, which are described by two-sided conditional probabilities. In such Markov Fields the time interpretation of past and future is being replaced by the space interpretation of an interior volume, surrounded by an exterior to the left and to the right. If we relax the Markov requirement to weak dependence, that is, continuous dependence, either on the past (generalising the Markov-Chain description) or on the external configuration (generalising the Markov-Field description), it turns out this equivalence breaks down, and neither class contains the other. In one direction this result has been known for a few years, in the opposite direction a counterexample was found recently. Our counterexample is based on the phenomenon of entropic repulsion in long-range Ising (or "Dyson") models.