Liouville Measures with Finite Entropy

• Liouville measures with finite entropy are probability measures on groups or vertex-transitive graphs where every bounded harmonic function is constant and the Shannon entropy remains finite.
• They reveal a precise connection between the decay of return probabilities, heat kernel behavior, and entropy growth through techniques such as spectral isoperimetry and entropy decomposition.
• Examples like lamplighter groups and the infinite symmetric group illustrate striking phenomena including the non-convex stability of the Liouville property under convex combinations.

A Liouville measure with finite entropy is a probability measure on a countable group or vertex-transitive graph such that all bounded harmonic functions for the associated random walk are constant, and the Shannon entropy is finite. The interplay between return probabilities, entropy growth, and the structure of the Poisson boundary is central for such measures. Rigorous criteria for the Liouville property and sharp entropy bounds have been developed, revealing deep relations between heat kernel decay, spectral isoperimetry, random walks, and ergodic properties of groups. Notably, recent constructions highlight striking phenomena such as the non-stability of the Liouville property under convex combinations, even in the finite-entropy regime. The lamplighter group and the infinite symmetric group provide principal examples.

1. Fundamental Definitions and Characterizations

Let $G$ be a countable group and $\mu$ a symmetric (i.e., $\mu(g) = \mu(g^{-1})$), fully supported probability measure generating $G$ as a semigroup. The measure $\mu$ is said to have the Liouville property if every bounded $\mu$-harmonic function $f : G \to \mathbb{R}$,

$$f(g) = \sum_{h\in G} f(gh)\, \mu(h) \qquad \forall g\in G,$$

is constant. Equivalently, the Poisson boundary $\partial_P(G,\mu)$ is trivial, i.e., consists of a single point.

The finite entropy condition is

$$H(\mu) = -\sum_{g\in G} \mu(g) \log \mu(g) < \infty,$$

which ensures well-behaved ergodic and boundary phenomena for the induced random walk. Define the $n$-fold convolution $\mu^{*n}$ as the law of the position $W_n$ of the random walk after $n$ steps. The asymptotic entropy is

$$h(\mu) = \lim_{n\to\infty} \frac{1}{n} H(\mu^{*n}),$$

and for $H(\mu)<\infty$, the Liouville property is equivalent to $h(\mu)=0$ by the Derriennic–Kaimanovich–Vershik criterion (Forghani et al., 1 Feb 2026).

2. Entropy Growth, Heat Kernel Decay, and the Liouville Threshold

A crucial connection is established between the decay of return probabilities and the Liouville property. Let

$\mu^{(2n)}(e) = P(W_{2n}=e)$

be the 2$n$-step return probability. The following are central results (Peres et al., 2016):

• Liouville property via slow heat-kernel decay: If $-\log \mu^{(2n)}(e) = o(n^{1/2})$, then $(G,\mu)$ is Liouville. This criterion is sharp in the regime of sub-Gaussian return probability decay.
• Entropy bounds from polynomial-type decay: If $-\log \mu^{(2n)}(e) = O(n^\beta)$ for $\beta \in (0,1/2)$, then

$H(\mu^{(n)}) = O\left(n^{\beta/(1-\beta)}\right).$

The exponent $\beta/(1-\beta)$ is optimal up to logarithmic corrections, as demonstrated via permutational wreath products over bubble graphs.

The methods rely on analyzing the spectral profile $\Lambda_\mu(v)$ (minimum Dirichlet eigenvalue over sets of size at most $v$), leveraging the Coulhon–Grigor'yan relation between return probabilities and isoperimetry, and applying entropy decomposition relative to nested test sets.

3. Extension to Transitive Graphs and Group Actions

These results generalize to random walks on transitive graphs. For a vertex-transitive, locally finite graph $X$ and symmetric, $\Gamma$-invariant Markov kernel $P$ of finite entropy on the automorphism group $\Gamma$, one lifts $P$ to a symmetric measure on $\Gamma$. Similar return probability and entropy conclusions hold with $\mu^{(2n)}(e)$ replaced by $P^{2n}(o,o)$ and $H(\mu^{(n)})$ by the entropy $H_P(n)$ of the walk on $X$ (Peres et al., 2016).

4. Classification and Examples: Finitely Liouville Groups

The only known non-trivial countable groups admitting Liouville measures of finite entropy are the lamplighter groups (e.g., $\mathbb{Z}_2 \wr \mathbb{Z}$, $\mathbb{Z}_2 \wr \mathbb{Z}^2$) and the infinite symmetric group $S_\infty$ (permutations of $\mathbb{N}$ with finite support). For these groups, one can construct measures of the form

$$\mu_i = t_i \sum_n p_n^{A_i} U(S_n) + (1-t_i) \sum_n q_n U(F_n),$$

where $U(\cdot)$ denotes the uniform law on a finite set, $F_n$ are Følner sets, $S_n$ are large “switching” sets, and $p,q$ are infinite probability distributions on $\mathbb{N}$ with $H(p),H(q)<\infty$. Such $\mu_i$ have $H(\mu_i)<\infty$ and induce random walks whose bounded harmonic functions are constant (Forghani et al., 1 Feb 2026).

5. Non-Convexity and Instability Phenomena

A fundamental discovery is the failure of convex stability for the set of Liouville measures with finite entropy on certain amenable groups. For non-hyper-FC-central, finitely Liouville countable groups, one can construct, for any $k\geq2$, symmetric fully supported measures $\mu_1,\ldots,\mu_k$ with $H(\mu_i)<\infty$ such that

$\text{$\mu = \sum_{i=1}^k p_i\,\mu_i$ is Liouville} \quad \iff \quad |\{i:\, p_i>0\}| < k$

(Forghani et al., 1 Feb 2026). When $k$ or more $\mu_i$ participate in a convex combination, the Poisson boundary becomes non-trivial. The mechanism employs partitioning $\mathbb{N}$ into “$k$-covers,” coupling to record times, and balancing long-range mixing with intervals of regular (Følner) mixing. The entropy control relies on the continuity properties of entropy in a suitable metric $$D(\theta,\lambda) = \|\theta-\lambda\|_1 + |H(\theta)-H(\lambda)|$$.

This non-convexity directly answers a question of Kaimanovich and implies that, in contrast to many ergodic or geometric properties, Liouville measures with finite entropy do not form a convex subset.

6. Open Problems and Critical Regimes

Several directions remain unresolved:

• It is unknown whether any other non-trivial (i.e., beyond lamplacters and $S_\infty$) ICC amenable groups admit Liouville measures with finite entropy (Forghani et al., 1 Feb 2026).
• For the critical regime where $-\log \mu^{(2n)}(e) \simeq n^{1/2}$ (e.g., for $\mathbb{Z}_2 \wr \mathbb{Z}^2$), the behaviour of the Liouville property is subtle; it is open whether return probabilities decaying at the rate $e^{-n^{1/2}}$ guarantee Liouville (Peres et al., 2016).
• The sharp boundary between entropy growth, boundary triviality, and return probability persists as a central issue for random walks on groups and graphs.

7. Summary Table: Key Results

Framework	Criterion	Consequence
$(G,\mu)$ symmetric, $H(\mu)<\infty$	$-\log \mu^{(2n)}(e) = o(n^{1/2})$	Liouville: all bounded harmonic functions constant (Peres et al., 2016)
$(G,\mu)$ symmetric, $H(\mu)<\infty$	$-\log \mu^{(2n)}(e) = O(n^\beta)$, $\beta<1/2$	$H(\mu^{(n)})=O(n^{\beta/(1-\beta)})$
Convex $\sum p_i\mu_i$, finite entropy	$| \{i: p_i>0\}| < k$	Liouville; otherwise, non-Liouville if $| \{i: p_i>0\}| \geq k$ (Forghani et al., 1 Feb 2026)

The study of Liouville measures with finite entropy thus exposes intricate boundaries between random walk asymptotics, entropy, and the geometry of harmonic functions, while highlighting sharp dichotomies and non-closure phenomena in the space of such measures.