Poisson representations for tree-indexed Markov chains

TL;DR

This work analyzes when tree-indexed Markov chains admit a Poisson representation, extending the class ${\mathcal R}$ from linear graphs to trees. It proves a sharp phase transition on finite trees: for a finite tree with boundary size $k$ and maximum degree $m$, there are thresholds $r_0(k)$ and $r_1(m)$ such that representability depends on both $r$ and the regime of $p$ (near 0 or near 1). It further shows that certain infinite trees, notably octopus trees $T_m$, admit Poisson representations for some parameter ranges, with a $p$-independent threshold $r_2(m)$ and a special case $m=3$ where $r_2(3)=r_1(3)=1/2$, making representability determined by $r$ alone in that case. The paper also strengthens and broadens prior results by providing streamlined proofs, signed-measure extensions, and scaling arguments that connect finite and infinite-tree behavior, clarifying when Ising-like models fail to be Poisson representable and enriching the theory of tree-indexed processes.

Abstract

In~\cite{fgs}, the class of Poisson representable processes was introduced. Several well-known processes were shown not to belong to this class, with examples including both the Curie Weiss model and the Ising model on $ \mathbb{Z}^2 $ for certain choices of parameters. Curiously, it was also shown that all positively associated $ \{ 0,1 \}$-valued Markov chains do belong to this class. In this paper, we interpolate between Markov chains and Ising models by considering tree-indexed Markov chains. In particular, we show that for any finite tree that is not a path, whether or not the corresponding tree-indexed Markov chain is representable always depends on the parameters. Moreover, we give an example of a family of infinite trees such that the corresponding tree-indexed Markov chains are representable for some non-trivial parameters. In addition, we give alternative proofs and arguments and also strengthen several of the results in~\cite{fgs}.

Figures (7)

• Figure 1: The octopus trees $T_3,$$T_4,$ and $T_5.$
• Figure 2:
• Figure 3: The figures above illustrate the settings of Lemma \ref{['lemma: observation 1']}, Lemma \ref{['lemma: observation 2']} and Lemma \ref{['lemma: observation 3']}. In all figures, we have drawn a finite tree $T$, a set $S$ in black, the corresponding set $\mathcal{B}^+_T(S)$ in white, a set $E$ in red, and $R_{T,S}$ in red.
• Figure 4: The settings of \ref{['item: case 1']}, \ref{['item: case 2']}, and \ref{['item: case 3']} in the proof of Proposition \ref{['proposition: 2/3 trees']}. In the figures, the set $L$ is drawn in black, and a set $E$ in red.
• Figure 5: The setting of the end of the proof of Proposition \ref{['proposition: 2/3 trees']}, illustrating the bijection between $\hat{P}(E(T_S))$ and the direct product of $\hat{P}(E(T_{S\smallsetminus L}))$ and $\hat{P}(E_v).$

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 2.1: Lemma 2.12 in fgs
• Lemma 2.2: Lemma 2.14 in fgs
• Lemma 3.1
• proof
• Proposition 3.2
• proof