Non-stability of Liouville measures under convex combinations

For every non-hyper-FC-central countable amenable group and every $k\geq 2$, we provide a sequence of symmetric, fully supported probability measures such that their convex combination is non-Liouville (that is it admits a non-constant bounded harmonic function, equivalently, the Poisson boundary is non-trivial) if and only if at least $k$ of them appear in the convex combination. Particularly, our result implies that the set of Liouville measures is not closed under convex combination, which answers a question of Kaimanovich. We also provide a similar result under the additional assumption of finite entropy for those non-hyper-FC-central countable groups with the property that every symmetric, finitely supported probability measure is Liouville. These groups are the only known non-trivial examples of countable groups that admit Liouville measures with finite entropy. Examples include the lamplighter group over $\mathbb{Z}$ and $\mathbb{Z}^2$, and the infinite symmetric group of finite permutations on $\mathbb{Z}$.

💡 Research Summary

The paper addresses a fundamental question in the theory of random walks on groups: whether the set of Liouville probability measures on a given group is closed under finite convex combinations. A probability measure μ on a countable group G is called Liouville if every bounded μ‑harmonic function is constant, equivalently if the Poisson boundary of the μ‑random walk is trivial. While it is known that abelian, nilpotent and more generally Choquet‑Deny groups are Liouville for all measures, and that groups with an ICC quotient (i.e., non‑hyper‑FC‑central groups) admit non‑Liouville measures, the stability of the Liouville property under convex combinations remained open for amenable non‑hyper‑FC‑central groups. This question was explicitly posed by Kaimanovich.

The authors give a negative answer. For any countable amenable group G that is not hyper‑FC‑central and for any integer k ≥ 2, they construct a sequence {μ_i} of symmetric, fully supported probability measures such that:

• Any convex combination involving at most k − 1 of the μ_i’s is Liouville.
• Every convex combination that uses k or more of the μ_i’s (finite or infinite) is non‑Liouville.

Thus the Liouville set is not convex‑closed. The construction intertwines two classical tools:

1. Følner sets, which exist precisely because G is amenable. Random walks built from Følner sets (à la Kaimanovich‑Vershik) are Liouville but typically have infinite entropy.
2. Switching sets, introduced in earlier work of the authors, which yield non‑Liouville measures by forcing the random walk to “switch” between far‑apart regions of the group.

The novelty lies in the delicate probabilistic framework that decides when a convex combination inherits the Liouville property. The authors introduce long gaps and fit times for probability measures on ℕ. A long gap for a mixture α = t p + (1−t) q (with q fully supported) means that infinitely often the contribution from p is dominated by that from q at a given integer m. Fit times are special record times where a value m appears for the first time under the q‑part while the p‑part has not yet produced a larger value. They prove that if a mixture has a long gap, then the associated sequence of i.i.d. variables possesses infinitely many fit times. This combinatorial structure is then lifted to random walks on G, allowing the authors to control the Poisson boundary.

A key combinatorial device is a k‑cover of ℕ: a family of subsets {A_j} such that any k distinct indices cover ℕ, while any k − 1 of them leave an infinite remainder. Using a k‑cover, the authors define measures α_j = t_j p_{A_j} + (1−t_j) q. They show that any convex combination of fewer than k of the α_j’s lacks a long gap (hence the associated random walk is Liouville), whereas any combination involving k or more does have a long gap, forcing non‑Liouville behavior via the Erschler‑Kaimanovich framework.

Beyond the infinite‑entropy setting, the paper also treats the case of finite entropy. Certain amenable groups—called finitely Liouville—have the property that every symmetric finitely supported measure is Liouville. Known examples include lamplighter groups over ℤ and ℤ², and the infinite symmetric group of finite permutations of ℤ. For such groups, the authors prove an analogue of the main theorem: there exist symmetric, fully supported measures with finite entropy satisfying the same k‑threshold phenomenon. The proof relies on the equivalence (Derriennic, Kaimanovich‑Vershik) that a finite‑entropy measure is Liouville iff its asymptotic entropy is zero. The authors introduce a D‑metric, a distance on probability measures that makes the map μ ↦ h(μ^{*n}) (entropy of the n‑fold convolution) continuous. Using this metric they approximate fully supported finite‑entropy measures by finitely supported ones, where the finitely Liouville hypothesis guarantees zero asymptotic entropy for any convex combination of fewer than k measures. When k or more measures are combined, the D‑metric shows a positive jump in entropy, yielding non‑Liouville behavior.

The paper’s structure is as follows:

1. Section 2 develops the record‑time machinery on ℕ, defines ladders, long gaps, k‑covers, and proves that long gaps imply infinitely many fit times.
2. Section 3 recalls Følner and switching sets, and the standard construction of non‑Liouville measures for ICC groups.
3. Section 4 builds the Liouville measures using long gaps and proves Theorem A (the infinite‑entropy version).
4. Section 5 reviews entropy basics, defines the D‑metric, and proves Theorem B (the finite‑entropy version).

The results have several noteworthy implications:

• They settle Kaimanovich’s convex‑closure question negatively for a broad class of amenable groups.
• They exhibit a “secret‑sharing” phenomenon: only when at least k participants (measures) collaborate does the combined system reveal non‑trivial boundary information.
• The introduction of long gaps, fit times, and the D‑metric provides new tools that may be useful in other problems concerning Poisson boundaries, entropy, and random walks on groups.

In summary, the authors demonstrate that Liouville measures are highly non‑stable under convex combinations in non‑hyper‑FC‑central amenable groups, both in the infinite‑entropy and finite‑entropy regimes, thereby deepening our understanding of the interplay between group structure, random walk behavior, and harmonic analysis.