Curvature of basis exchange walks

TL;DR

This work studies the discrete Ollivier-Ricci curvature of basis exchange walks on matroids, addressing whether these walks are universally non-negatively curved. The authors develop a down-step coupling to establish a general lower bound on curvature and use a distance-based analysis to obtain an upper bound, culminating in tight bounds under various regimes. They prove a universal lower bound $\bkappa \ge -1 + \frac{2}{k} + \frac{3(k-1)}{k(n-k+1)}$ (and $\,\u00bkappa \ge \frac{1}{k}$ when $n=k+1$) and derive an explicit upper bound involving the sets $J$, $A_u$, and neighborhood sizes; these tools enable the construction of matroids with both non-negative and negative curvature. The paper provides concrete examples, including rank-2, uniform, Vámos, and graphic matroids, to demonstrate the existence of negative curvature (e.g., a path-based construction in a $K_6$ graphic matroid yields $\,\kappa \le -\frac{2}{25}$). Overall, the results show that non-negative curvature is a delicate property not guaranteed by stationary distribution or representability, shedding light on the relationship between curvature and spectral independence in basis exchange walks.

Abstract

We prove both lower and upper bounds on the Ollivier-Ricci curvature of the basis exchange walk on a matroid. We give several examples of non-negatively curved basis exchange walks and negatively curved basis exchange walks.

Figures (3)

• Figure 1: The down-step coupling for two neighboring states in the graphic matroid induced by $K_4$.
• Figure 2: Dependent sets in the Vámos matroid.
• Figure 3: Labeling of outer edges of $K_6$.

Theorems & Definitions (31)

• Definition 2.1: Matroid
• Definition 2.2: Rank
• Definition 2.3: Graphic matroids
• Definition 2.4: Down-up walk
• Definition 2.5: Wasserstein distance
• Definition 2.6: Ollivier--Ricci curvature
• Definition 3.1
• Definition 3.2: Down-step coupling
• Remark 1
• Theorem 3.1