Uniform even subgraphs and graphical representations of Ising as factors of i.i.d

Abstract

We prove that the Loop O(1) model, a well-known graphical expansion of the Ising model, is a factor of i.i.d. on unimodular random rooted graphs under various conditions, including in the presence of a non-negative external field. As an application we show that the gradient of the free Ising model is a factor of i.i.d. on unimodular planar maps having a locally finite dual. The key idea is to develop an appropriate theory of local limits of uniform even subgraphs with various boundary conditions and prove that they can be sampled as a factor of i.i.d. Another key tool we prove and exploit is that the wired uniform spanning tree on a unimodular transient graph is a factor of i.i.d. This partially answers some questions posed by Hutchcroft [32].

A factor of i.i.d. with uniform marginals and infinite clusters spanned by equal labels

“…We will show that an FIID process which witnesses the truth of Theorem 1.1 cannot be finitary. It was raised as an open question if there exists an FIID process on a non-amenable group which is not finitary during the discussion of a talk by Ray and Spinka (to access the talk, see [RS22], the talk is based on the paper [ARS21]), and our construction thus gives an affirmative answer to this question. The label set in our process is the interval [0, 1], and to our knowledge, the question if there exists a non-finitary FIID with a finite label set on a non-amenable group is still open.…”