Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generic sampling and invariant measures on the space of $k$-uniform hypergraphs

Nathanael Ackerman, Cameron Freer, Kyle Gannon, James E. Hanson, Rehana Patel

TL;DR

The paper develops a model-theoretic framework for representing ergodic, Sym(ℕ)-invariant measures on spaces of countable k-uniform hypergraphs through generic sampling of Keisler measures. By constructing, for any Borel k-ary hypergraphon W, an excellent global Keisler measure μ_W whose Morley-product-based generic sampling reproduces the hypergraphon-generated invariant measure, it unifies probabilistic invariant measures with model-theoretic sampling. The main contributions include a graphon (k=2) and hypergraphon representation theorem, a 0–1 law for excellent measures, and a detailed analysis of how graphon/hypergraphon sampling corresponds to generic sampling over monster models of the Fraïssé limit (the Rado graph for graphs). This bridges exchangeable structure representations with model-theoretic constructions, yielding new, tame examples of Keisler measures outside NIP contexts and extending representation results to hypergraphs. The results provide a canonical, scalable method to realize all ergodic invariant measures on countable k-uniform hypergraphs via Morley-product generic sampling, with implications for both combinatorics and logical foundations of probability.

Abstract

We prove a model-theoretic representation theorem for the distribution of an ergodic exchangeable $k$-uniform hypergraph: every such measure arises as the pushforward of the countably-iterated Morley product of a global Borel-definable Keisler measure over the countable universal homogeneous $k$-uniform hypergraph. We show this by starting with a Borel $k$-hypergraphon $W$ and constructing a Keisler measure $μ_{W}$ such that generic sampling with respect to $μ_{W}$ yields the same invariant measure as does the standard hypergraphon sampling procedure with respect to $W$. When $k = 2$, our results give a new representation theorem for ergodic exchangeable graphs via Keisler measures over a monster model of the Rado graph.

Generic sampling and invariant measures on the space of $k$-uniform hypergraphs

TL;DR

The paper develops a model-theoretic framework for representing ergodic, Sym(ℕ)-invariant measures on spaces of countable k-uniform hypergraphs through generic sampling of Keisler measures. By constructing, for any Borel k-ary hypergraphon W, an excellent global Keisler measure μ_W whose Morley-product-based generic sampling reproduces the hypergraphon-generated invariant measure, it unifies probabilistic invariant measures with model-theoretic sampling. The main contributions include a graphon (k=2) and hypergraphon representation theorem, a 0–1 law for excellent measures, and a detailed analysis of how graphon/hypergraphon sampling corresponds to generic sampling over monster models of the Fraïssé limit (the Rado graph for graphs). This bridges exchangeable structure representations with model-theoretic constructions, yielding new, tame examples of Keisler measures outside NIP contexts and extending representation results to hypergraphs. The results provide a canonical, scalable method to realize all ergodic invariant measures on countable k-uniform hypergraphs via Morley-product generic sampling, with implications for both combinatorics and logical foundations of probability.

Abstract

We prove a model-theoretic representation theorem for the distribution of an ergodic exchangeable -uniform hypergraph: every such measure arises as the pushforward of the countably-iterated Morley product of a global Borel-definable Keisler measure over the countable universal homogeneous -uniform hypergraph. We show this by starting with a Borel -hypergraphon and constructing a Keisler measure such that generic sampling with respect to yields the same invariant measure as does the standard hypergraphon sampling procedure with respect to . When , our results give a new representation theorem for ergodic exchangeable graphs via Keisler measures over a monster model of the Rado graph.
Paper Structure (11 sections, 29 theorems, 125 equations)

This paper contains 11 sections, 29 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Keisler measures
  5. Space of labeled L-structures
  6. 0-1 law for excellent measures
  7. Graphons, generic sampling, and the Rado graph
  8. $\emptyset$-Invariant measures
  9. Graphon representation theorem
  10. Hypergraphons, generic sampling, and random hypergraphs
  11. Proof of \ref{['prop:nightmare']}

Key Result

Proposition 3.6

Let $\mu \in \mathfrak{M}^{\mathrm{inv}}_{x}(\mathcal{U},M)$, and suppose that $\mu$ does not concentrate on points and $\mu$ is $M$-excellent. Then $\mathbb{P}_{\mu}(\mathbb{B}_{N}) \in \{0,1\}$ for any $\mathcal{L}$-structure $N$.

Theorems & Definitions (89)

  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • Definition 2.12
  • ...and 79 more