# Curvature-dimension inequalities for non-local operators in the discrete setting

## TL;DR

This paper investigates curvature-dimension inequalities for non-local operators on discrete lattices, revealing conditions under which these operators have finite or infinite dimension and analyzing their curvature properties.

## Contribution

It provides new results on when non-local operators on lattices have finite or infinite dimension and explores curvature properties related to fractional Laplacians and sparse kernels.

## Key findings

- Operators with finite second moment have finite dimension.
- Operators related to fractional Laplacians often lack finite dimension.
- Many operators with sparse support have no positive curvature.

## Abstract

We study Bakry-\'Emery curvature-dimension inequalities for non-local operators on the one-dimensional lattice and prove that operators with finite second moment have finite dimension. Moreover, we show that a class of operators related to the fractional Laplacian fails to have finite dimension and establish both positive and negative results for operators with sparsely supported kernels. Moreover, a large class of operators is shown to have no positive curvature. The results correspond to CD inequalities on locally infinite graphs.