Intertwining curvature bounds for well-studied graph models

Determine explicit lower bounds on the intertwining curvature for graphs modeling interacting particle systems and for graphs associated with random transpositions on the symmetric group, where Bakry–Émery and entropic Ricci curvature bounds are already established. Formally, for each such graph, construct a strongly continuous semigroup on edge functions that intertwines with the heat semigroup and proves an intertwining curvature lower bound K as defined for finite weighted graphs, yielding corresponding gradient estimates for all operator mean functions.

Background

The paper introduces intertwining curvature as a new curvature notion for graphs (and quantum Markov semigroups) that is stronger than both Bakry–Émery and entropic Ricci curvature and implies the full suite of analytic consequences from both frameworks. Establishing intertwining curvature bounds can be substantially simpler than proving entropic curvature bounds directly.

While the paper provides general theory, universal bounds, and several classes of examples (e.g., complete graphs and mapping representations), the authors highlight that few graph examples have been explored so far for intertwining curvature. For several important graph models, such as interacting particle systems and the random transposition walk on the symmetric group, Bakry–Émery and entropic curvature bounds are known, but intertwining curvature bounds have not yet been developed.

Resolving this problem would bridge existing curvature results and potentially yield sharper functional inequalities and gradient estimates for these well-studied models via the stronger intertwining curvature framework.