Exchangeable interval hypergraphs and limits of ordered discrete structures

Abstract

A hypergraph $(V,E)$ is called an interval hypergraph if there exists a linear order $l$ on $V$ such that every edge $e\in E$ is an interval w.r.t. $l$; we also assume that $\{j\}\in E$ for every $j\in V$. Our main result is a de Finetti-type representation of random exchangeable interval hypergraphs on $\mathbb{N}$ (EIHs): the law of every EIH can be obtained by sampling from some random compact subset $K$ of the triangle $\{(x,y):0\leq x\leq y\leq1\}$ at i.i.d. uniform positions $U_{1},U_{2},\dots$, in the sense that, restricted to the node set $[n]:=\{1,\dots,n\}$ every nonsingleton edge is of the form $e=\{i\in[n]:x<U_{i}<y\}$ for some $(x,y)\in K$. We obtain this result via the study of a related class of stochastic objects: erased-interval processes (EIPs). These are certain transient Markov chains $(I_{n},\eta_{n})_{n\in\mathbb{N}}$ such that $I_{n}$ is an interval hypergraph on $V=[n]$ w.r.t. the usual linear order (called interval system). We present an almost sure representation result for EIPs. Attached to each transient Markov chain is the notion of Martin boundary. The points in the boundary of EIPs can be seen as limits of growing interval systems. We obtain a one-to-one correspondence between these limits and compact subsets $K$ of the triangle with $(x,x)\in K$ for all $x\in[0,1]$. Interval hypergraphs are a generalizations of hierarchies and as a consequence we obtain a representation result for exchangeable hierarchies, which is close to a result of Forman, Haulk and Pitman in (Probab. Theory Related Fields 172 (2018) 1–29). Several ordered discrete structures can be seen as interval systems with additional properties, that is, Schröder trees (rooted, ordered, no node has outdegree one) or even more special: binary trees. We describe limits of Schröder trees as certain tree-like compact sets. These can be seen as an ordered counterpart to real trees, which are widely used to describe limits of discrete unordered trees. Considering binary trees, we thus obtain a homeomorphic description of the Martin boundary of Rémy’s tree growth chain, which has been analyzed by Evans, Grübel and Wakolbinger in (Ann. Probab. 45 (2017) 225–277).