Aldous spectral-gap conjecture for interchange processes on weighted hypergraphs

Prove that the spectral gap of the interchange process on every arbitrary weighted hypergraph is attained by a one-particle eigenfunction, thereby resolving Caputo’s conjecture for weighted-hypergraph interchange processes.

Background

The paper places its results within the broader study of Aldous-type spectral-gap phenomena, in which the dominant relaxation mode is represented by a low-particle observable. The classical Aldous conjecture for interchange processes on weighted graphs is known, but its proposed extension to arbitrary weighted hypergraphs remains unresolved.

The authors distinguish this interchange-process problem from the stochastic exchange models treated in the paper: their degree-two reduction theorem does not establish the one-particle reduction conjectured for weighted-hypergraph interchange processes. The problem is therefore a genuinely unresolved extension of the graph case.