Lattice random-field Widom--Rowlinson models

TL;DR

This paper investigates the phase transition behavior of the Widom--Rowlinson model on integer lattices under random fields, revealing dimension-dependent effects and extending known results from the random-field Ising model.

Contribution

It extends the understanding of phase transitions in the Widom--Rowlinson model under random fields, especially in low and high dimensions, using contour methods.

Findings

No phase transition in dimensions d ≤ 2 under any non-trivial random field.

Phase transition persists in dimensions d ≥ 3 for Gaussian random fields at high densities.

The results generalize the classical random-field Ising model findings to the Widom--Rowlinson model.

Abstract

We consider the Widom--Rowlinson model on $\mathbb{Z}^d$ subject to a symmetric i.i.d.\ random field. We prove that for dimensions $d\le 2$ any non-trivial random field leads to an absence of a phase transition. In contrast, in dimensions $d\ge 3$ and for Gaussian random fields, phase-transition behavior of the model is maintained for sufficiently large densities of occupied sites. This extends the general picture known from the classical random-field Ising model to the random-field Widom--Rowlinson model. Following the general proof route of Aizenman--Wehr as well as Ding--Zhuang, our main contribution rests on adequate notions of contours and their associated generalized spin-flip operation to deal with hard-core repulsions.