Isomorphisms between random $d$-hypergraphs

TL;DR

This paper studies the largest common induced subhypergraph between two independent random uniform hypergraphs, revealing a phase transition and asymptotic concentration, extending known graph results to hypergraphs.

Contribution

It generalizes existing graph results to hypergraphs, characterizing the distribution and phase transition of the largest common induced subhypergraph.

Findings

Distribution concentrates on two points asymptotically

Identifies a phase transition for hypergraph inclusion

Extends results from graphs to hypergraphs

Abstract

We characterize the size of the largest common induced subgraph of two independent random uniform $d$-hypergraphs of different sizes with $d\geq 3$. More precisely, its distribution is asymptotically concentrated on two points, and we obtain as a consequence a phase transition for the inclusion of the smallest hypergraph in the largest one. This generalizes to uniform random $d$-hypergraphs the results of Chatterjee and Diaconis for uniform random graphs. Our proofs rely on the first and second moment methods.