Curvature of basis exchange walks

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.

💡 Research Summary

This paper investigates the Ollivier‑Ricci curvature of the basis‑exchange walk, a natural Markov chain defined on the bases of a matroid. The authors first recall the definition of a matroid and the down‑up (basis‑exchange) walk, which proceeds by uniformly dropping an element from the current basis and then uniformly adding an element that restores the basis property. The stationary distribution of this walk is uniform over all bases.

Ollivier‑Ricci curvature κ of a Markov chain P on a metric space (X,d) is defined via the Wasserstein distance: for any two states S,T∈X, the distance between the one‑step transition distributions satisfies W(P(S,·),P(T,·)) ≤ (1−κ)·d(S,T). When d is the graph distance induced by the chain, it suffices to check adjacent states.

Lower bound (Theorem 3.1).
The authors construct a specific coupling, called the “down‑step coupling”, for two adjacent bases S={s,u₁,…,u_{k−1}} and T={t,u₁,…,u_{k−1}}. The coupling first matches the down‑step (the element that is dropped) and then tries to match the up‑step (the element that is added). By analysing the sets N(B−u) of admissible insertions after a given deletion, they obtain an explicit bound on the expected distance after one step:

• If the deleted element is the distinguished one (s vs. t), the two walks coalesce (distance 0).
• If a common element u_i is deleted, two cases arise depending on whether the other distinguished element belongs to the insertion set. In the worst case the expected distance is at most 2−3/(n−k+1).

Averaging over all possible deletions yields the general lower bound
κ ≥ −1 + 2/k + 3(k−1)k/(n−k+1) for n>k+1, and κ ≥ 1/k when n=k+1. This shows that curvature is always bounded away from −1, and for small rank or small ground set the bound becomes positive.

Upper bound (Theorem 5.1).
To obtain an upper bound, the authors consider any coupling and lower‑bound the probability that the distance after one step is at least 2. They introduce the set J={u : t∈N(S−u), u≠s} and, for each u∈J, the asymmetric set A_u = (N(S−u) {t}) \ N(T−u). Roughly, A_u consists of insertions available from S but not from T after deleting u. Using these definitions they prove
κ ≤ 1/k + (1/k)∑_{u∈J} ( 1/|N(T−u)| − |A_u|/|N(S−u)| ).
When the insertion neighborhoods of the two bases are highly overlapping, the sum is small and κ is close to the lower bound; when the neighborhoods are very different, the sum can be large enough to make κ negative.

Concrete examples.

• Rank‑2 matroids: Any rank‑2 matroid has positive curvature (Corollary 4.1) because adjacent bases differ by exactly one element.
• Small ground sets: For n≤7, every matroid has non‑negative curvature (Corollary 4.2).
• Small graphic matroids: Graphic matroids of graphs with ≤4 vertices are non‑negatively curved (Corollary 4.3).
• Uniform matroids: For the uniform matroid U_{k,n}, the lower bound simplifies to κ ≥ 1 − (k−1)(n−k)/(k(n−k+1)). When n is sufficiently larger than k, this is positive (Lemma 4.4).
• Vámos matroid: This non‑representable rank‑4 matroid on 8 elements contains 65 of the 70 possible 4‑element bases. By a careful case analysis of the down‑step coupling, the authors show that the expected distance never exceeds 1, yielding κ>0 (Lemma 4.5).
• Graphic matroids with edge‑disjoint cycles: If a graph’s cycles are edge‑disjoint, the basis‑exchange walk on its graphic matroid is positively curved (Lemma 4.6). The proof exploits the fact that each cycle can be treated independently, allowing a coupling that keeps the expected distance at most 1 (often ½).

Negative curvature instances.
Using the upper bound, the authors identify configurations where J is large and many asymmetric insertions exist, making the sum in Theorem 5.1 dominate the 1/k term. They construct explicit matroids (e.g., certain graphs with overlapping cycles) where κ becomes negative, demonstrating that non‑negative curvature is not guaranteed even though the walk enjoys other nice properties such as a bounded spectral gap and strong log‑concavity of the stationary distribution.

Implications.
Positive Ollivier‑Ricci curvature implies several powerful geometric and probabilistic results: a discrete Bonnet‑Myers theorem (diameter bound), lower bounds on the spectral gap, and mixing‑time guarantees via path coupling. The paper shows that while basis‑exchange walks are uniformly 1‑spectrally independent (a property linked to strong concentration), this does not force non‑negative curvature. Consequently, spectral independence and Ollivier‑Ricci curvature are distinct notions of expansion; one does not imply the other. This resolves an open question raised by recent works (