A note on spatial monotonicity for one-dimensional spin systems

TL;DR

This paper demonstrates a fundamental property of one-dimensional spin systems with attractive, translation-invariant interactions, showing that configurations evolve in a stochastically decreasing manner from a point onward over time.

Contribution

It introduces a stochastic domination argument to establish spatial monotonicity in one-dimensional spin systems with specific interaction properties.

Findings

Configurations are stochastically decreasing from a point over time.

The property holds for systems starting from the unit step function.

The proof simplifies the understanding of spatial monotonicity in such systems.

Abstract

We show that attractive, translation invariant, one-dimensional spin systems started from the unit step function possess the following basic property. At any time, the entire configuration from a point onward is stochastically decreasing with respect to distance from this point to the origin. The proof relies on a stochastic domination argument which exploits the interaction assumptions in a simple manner.