Every Salami has two ends

TL;DR

This paper characterizes 'salami' graphs with non-negative Ollivier Ricci curvature, proving they have exactly two ends, are recurrent, and closely resemble a line, with finite-dimensional harmonic function spaces.

Contribution

It establishes fundamental geometric and analytic properties of salami graphs, including their end structure, recurrence, and harmonic function characteristics, under curvature and weight conditions.

Findings

Salamis have exactly two ends.

Salamis are recurrent and quasi-isometric to a line.

Finite-dimensional space of harmonic functions.

Abstract

A salami is a connected, locally finite, weighted graph with non-negative Ollivier Ricci curvature and at least two ends of infinite volume. We show that every salami has exactly two ends and no vertices with positive curvature. We moreover show that every salami is recurrent and admits harmonic functions with constant gradient. The proofs are based on extremal Lipschitz extensions, a variational principle and the study of harmonic functions. Assuming a lower bound on the edge weight, we prove that salamis are quasi-isometric to the line, that the space of all harmonic functions has finite dimension, and that the space of subexponentially growing harmonic functions is two-dimensional. Moreover, we give a Cheng-Yau gradient estimate for harmonic functions on balls.