[Paper Review] Uniform even subgraphs and graphical representations of Ising as factors of i.i.d

Omer Angel, Gourab Ray+1 more|arXiv (Cornell University)|Dec 6, 2021

Markov Chains and Monte Carlo Methods4 citations

TL;DR

This paper establishes that the Loop $O(1)$ model, a key graphical representation of the Ising model, is a factor of i.i.d. on unimodular random rooted graphs under various conditions, including with a non-negative external field. The main result shows that the gradient of the free Ising model is a factor of i.i.d. on unimodular planar maps with locally finite duals, achieved via local limit theory of uniform even subgraphs and the construction of factors of i.i.d. for the wired uniform spanning tree.

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.

Motivation & Objective

• To determine whether the Loop $O(1)$ model on infinite graphs can be represented as a factor of i.i.d., a key question in stochastic processes and statistical mechanics.
• To extend the theory of factors of i.i.d. to uniform even subgraphs and the wired uniform spanning tree (WUSF) on unimodular random rooted graphs.
• To resolve open questions posed by Hutchcroft regarding the continuity of phase transitions in Ising models on nonamenable groups via factor of i.i.d. representations.
• To establish conditions under which the free and wired Loop $O(1)$ measures on infinite graphs arise as graph factors of i.i.d., particularly in planar and transient settings.
• To provide an alternative, explicit almost-sure proof of the convergence of Loop $O(1)$ measures on exhausting sequences, independent of Ising correlation function convergence.

Proposed method

• Develops a theory of local limits of uniform even subgraphs with various boundary conditions, showing they can be sampled as factors of i.i.d.
• Proves that the wired uniform spanning tree (WUSF) on a unimodular transient graph is a factor of i.i.d., using properties of unimodular random rooted graphs and stochastic domination.
• Uses the FK-Ising representation to relate the Loop $O(1)$ model to uniform even subgraphs of the random cluster model, leveraging duality and edge/vertex weight parameters.
• Applies Timár’s result on generating cycles in planar maps by finite-degree faces to construct a factor of i.i.d. for the free Loop $O(1)$ model on planar maps.
• Employs automorphism-equivariant measurable functions to define graph factors of i.i.d., ensuring invariance under graph automorphisms and measurability.
• Establishes that the free and wired Loop $O(1)$ measures on unimodular random rooted graphs are factors of i.i.d. under conditions such as planarity, local finiteness of the dual, and transience of FK-Ising clusters.

Experimental results

Research questions

• RQ1 Under what conditions is the Loop $O(1)$ model on an infinite unimodular random rooted graph a factor of i.i.d.?
• RQ2 Is the wired uniform spanning tree (WUSF) on a unimodular transient graph a factor of i.i.d.?
• RQ3 Can the gradient of the free Ising model be represented as a factor of i.i.d. on unimodular planar maps with locally finite duals?
• RQ4 Does the existence of finitely many geodesic cycles through each vertex in the FK-Ising model ensure that the Loop $O(1)$ model is a factor of i.i.d.?
• RQ5 For $x > 1$ or $y > 1$, are the free or wired Loop $O(1)$ measures still well-defined and factors of i.i.d. on infinite graphs?

Key findings

• The Loop $O(1)$ model is a factor of i.i.d. on unimodular random rooted graphs when the underlying graph is transient and the FK-Ising model has finitely many geodesic cycles through each vertex.
• The wired uniform spanning tree (WUSF) on a unimodular transient graph is a factor of i.i.d., resolving a partial case of a question posed by Hutchcroft.
• The gradient of the free Ising model is a factor of i.i.d. on unimodular planar maps with locally finite duals, as the free Loop $O(1)$ model on such maps is a factor of i.i.d.
• For planar maps, the collection of finite-degree faces generates the free cycle space, and assigning random parity to each face yields a factor of i.i.d. for the free Loop $O(1)$ model.
• In the amenable, vertex-transitive case, the free and wired Loop $O(1)$ measures coincide away from criticality, allowing the use of Theorem 1.1 to conclude the factor of i.i.d. property.
• For $x > 1$ or $y > 1$, the Loop $O(1)$ model can still be a factor of i.i.d. via duality: complementing open edges or vertices yields a model with parameters in $[0,1]$, to which the main theorems apply.