Non-uniqueness of phase transitions for graphical representations of the Ising model on tree-like graphs

TL;DR

This work demonstrates that percolation phase transitions for graphical representations of the Ising model on tree-like graphs can be nonunique and nonmonotone, constructing a graph $ ext{M}$ where the loop $O(1)$ and traced single random current exhibit disconnected percolation regimes. It establishes that, on the wired $d$-regular tree, the phase transitions for loop $O(1)$, single random current, and random-cluster not only exist but coincide, while free variants on graphs like $ ext{C}_n^d$ can display distinct thresholds. The paper also provides explicit critical points for these representations on $ ext{C}_n^d$ and the tree, and shows that the Bernoulli percolation threshold is not an obstruction for the uniform even subgraph percolation through edge-halving constructions and slightly supercritical clusters. Collectively, the results illuminate how long-range structure and cut-point factorisation drive nonmonotonic and nonunique percolation phenomena in graphical representations of Ising models on nonamenable, tree-like graphs, with implications for understanding phase transitions across couplings. The findings contribute to the broader theory of graphical representations by clarifying where standard monotonicity and uniqueness fail and where universality of wired-tree behavior holds.

Abstract

We consider the graphical representations of the Ising model on tree-like graphs. We construct a class of graphs on which the loop $\mathrm{O}(1)$ model and the single random current exhibit a non-unique phase transition with respect to the inverse temperature, highlighting the non-monotonicity of both models. It follows from the construction that there exist infinite graphs $\mathbb{G}\subseteq \mathbb{G}'$ such that the uniform even subgraph of $\mathbb{G}'$ percolates and the uniform even subgraph of $\mathbb{G}$ does not. We also show that on the wired $d$-regular tree, the phase transitions of the loop $\mathrm{O}(1)$, the single random current, and the random-cluster models are all unique and coincide.

Figures (5)

• Figure 1: The phase diagrams of the loop $\mathrm{O}(1)$, single random current, and random-cluster measures on the $d$-regular wired tree $\mathbb T^d$ coincide. The free measures on $\mathtt{C}_{n}^{d}$, the $d$-regular tree where every edge is substituted by a cycle, have different phase transitions. Finally, the free loop $\mathrm{O}(1)$ model on the monster $\mathbb{M}$ (constructed in the proof of Theorem \ref{['thm:non_uniqueness']}) has a non-unique phase transition. This is to be contrasted with the corresponding table for the hypercubic and hexagonal lattices in hansen2023uniform.
• Figure 2: The graph $G^{\diamond}$ (pictured to the right) along with its eight even subgraphs (including $G^{\diamond}$ itself). We let the outer paths be $n$ edges long and the inner paths be $m$ edges long. The nodes $a$ and $b$ are marked with dots. We list the number of edges of each subgraph, the corresponding weights and whether $a$ and $b$ are connected in the subgraph. (Sketch and text partially revised from klausen2021monotonicity.)
• Figure 3: Graph of the connection probability for the loop $\mathrm{O}(1)$ model on the graph $G^{\diamond}$, described in Figure \ref{['counter']}, for $n=12$ and $m=2$. See also klausen2021monotonicity for similar figures.
• Figure 4: An illustration of a part of the graph $\mathbb{M}$ built from the graphs $G^{\diamond}$ (see \ref{['counter']}) when $d=2$. To the left with the $d$-regular tree overlaid. To the right, the geometry of $\mathbb{M}$ at a single vertex of the initial tree.
• Figure 5: The critical $x_c$ on the graph $\mathtt{C}_{d,n}$ as a function of $d$ for the loop $\mathrm{O}(1)$, and double random current.

Theorems & Definitions (38)

• Theorem 1.1
• Corollary 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.2
• Definition 2.3
• Lemma 2.4: Cut point factorisation
• proof
• Lemma 2.5: Non-monotonicity of loop $O(1)$ two-point function
• proof