Random Walks on Free Products

• Random walks on free products are defined by combining groups into a structure that avoids consecutive identical factors, resulting in rich noncommutative dynamics.
• They exhibit analytic behaviors including central limit theorems, large deviations, and renewal structures that enable precise computation of drift, entropy, and variance.
• The topic bridges probabilistic, spectral, and geometric theories by revealing fractal boundaries, phase transitions, and sensitivity to transition parameters.

Random walks on free products are a central object of paper in probability on groups, spectral theory, geometric group theory, and analytic combinatorics. The free product construction combines several groups (or graphs, sets, or other discrete structures) into a new one whose elements have reduced words alternating between the factors without consecutive letters from the same factor, giving rise to a noncommutative and highly nonamenable structure. Random walks on such free products feature complex interactions between ergodic theory, renewal processes, generating functions, and fractal geometry, and exhibit a variety of phenomena such as central limit behaviors, large deviations, exotic local limit theorems, fractal boundaries, and analytic dependence of asymptotic quantities.

1. Structure of Free Products and Random Walks

Given a family of groups $\{G_i\}_{i=1}^r$ or graphs $\{V_i\}_{i=1}^r$, their free product $V = V_1 * V_2 * \cdots * V_r$ is defined as the collection of words $x_1 x_2 \cdots x_n$ where each $x_j \in V_{k_j} \setminus \{o_{k_j}\}$ and $k_j \neq k_{j+1}$, together with the root element $o$. The essential feature is the avoidance of consecutive letters from the same factor. Random walks on free products are constructed via convex combinations of transition kernels from each factor:

$$P = \alpha_1 \overline{P}_1 + \cdots + \alpha_r \overline{P}_r,$$

where $\overline{P}_i$ acts by appending a letter from $V_i$ (when allowed). Words thus evolve according to renewal dynamics, with the last cone entry times or block transitions forming the backbone for renewal theory in this context (Gilch, 2010, Gilch, 20 Oct 2025).

2. Asymptotic Escape Rates, Drift, and Entropy

The escape rate (drift) and the asymptotic entropy measure the speed and randomness of random walks in free products. For a position $X_n$ after $n$ steps,

• The drift $\lambda$ (graph distance) and block length drift $\ell$ are given by

$$\lambda = \lim_{n \to \infty}\frac{d(o, X_n)}{n}, \qquad \ell = \lim_{n \to \infty}\frac{\|X_n\|}{n}.$$

• The entropy takes the form

$$h = \lim_{n \to \infty} \frac{-1}{n} \log \pi_n(X_n),$$

where $\pi_n$ is the probability distribution of $X_n$ (Gilch, 2010, Gilch, 20 Oct 2025).

These limits exist under mild transience and mixing assumptions, and the rates can often be computed via expectations relative to the renewal increments arising from last cone entry times $T_k$:

$$\lambda = \frac{E[d(X_{T_0}, X_{T_1})]}{E[T_1 - T_0]}, \qquad \ell = \frac{\text{increment per block}}{E[T_1 - T_0]},$$

which hinges on the particular renewal structure of free products (Gilch, 20 Oct 2025).

The foundational result is that both $\lambda$ and $h$ vary real-analytically as functions of the model parameters (e.g. transition probabilities in finitely supported kernels), reflecting robust stability of random walk asymptotics under perturbations (Ledrappier, 2010, Gilch et al., 2015, Gilch, 20 Oct 2025).

3. Central Limit Theorems and Fluctuations

For random walks on free products or free products of graphs, central limit theorems (CLTs) for drift and entropy have been established (Gilch, 20 Oct 2025, Gilch, 2020, Gilch, 2022):

• For geometric drift,

$$\frac{d(o, X_n) - n\lambda}{\sigma_\lambda \sqrt{n}} \Rightarrow \mathcal{N}(0,1),$$

• For block length drift,

$$\frac{\|X_n\| - n\ell}{\sigma_\ell \sqrt{n}} \Rightarrow \mathcal{N}(0,1),$$

• For entropy,

$$\frac{-\log \pi_n(X_n) - n h}{\sigma_h \sqrt{n}} \Rightarrow \mathcal{N}(0,1).$$

Here, each variance $\sigma^2$ is computed as the expected squared deviation per renewal:

$$\sigma_\lambda^2 = \frac{E[(d(X_{T_0}, X_{T_1}) - (T_1-T_0)\lambda)^2]}{E[T_1 - T_0]},$$

and analogously for $\sigma_\ell^2$ and $\sigma_h^2$.

These variances also depend real-analytically on model parameters under finite support assumptions. For the entropy CLT, real-analyticity holds in the case of free products of finite graphs (Gilch, 20 Oct 2025, Gilch, 2020, Gilch, 2022).

4. Renewal Theory and Regeneration Structure

Analysis of random walks on free products heavily leverages their renewal structure. The sequence of last cone entry times $(T_k)$ partitions the trajectory into almost independent segments, enabling the application of strong laws, central limit theorems, and explicit computation of asymptotic characteristics (Gilch, 20 Oct 2025, Gilch, 2020, Gilch, 2022).

Expectations for increments between renewals (e.g., new distinct vertices, capacity additions) admit analytic formulas, often as convergent power series in parameters of the model (transition probabilities), due to the finite combinatorial complexity of finite-length paths and positive radius of convergence for the Green's functions involved.

5. Entropy, Rate of Escape, and Volume Growth: Inequalities and Connections

Random walks on free products exhibit inequalities connecting entropy, rate of escape, and geometric volume growth:

$$h \leq g_0 \cdot \ell_0 \qquad \text{and} \qquad h \leq g_1 \cdot \ell_1,$$

where $g_0$, $g_1$ are exponential growth rates with respect to block and natural graph metrics, $\ell_0$, $\ell_1$ are corresponding escape rates. These inequalities generalize Guivarc’h-type bounds and clarify how randomness and geometric expansion interact (Gilch, 2010, Gilch, 2010).

The entropy can also be calculated via generating function techniques and is equal to the rate of escape in a suitable "Greenian" metric:

$$h = \lim_{n\to\infty} -\frac{1}{n} \log F(o, X_n|1),$$

where $F$ is the first-visit generating function (Gilch, 2010).

6. Analyticity, Sensitivity, and Regularity

Drift, entropy, asymptotic range, and capacity associated with random walks on free products all vary real-analytically with respect to finitely supported transition parameters (Gilch et al., 2015, Ledrappier, 2010, Gilch, 2020, Gilch, 2022, Gilch, 20 Oct 2025). Such analytic dependence results from expressions for expectations and variances being finite sums of monomials (or ratios thereof) in the parameters, under positive radius of convergence ensured by transience.

This real-analyticity enables sensitivity analysis, phase transition studies, and parameter optimization, distinguishing free product random walks from those on abelian groups, where regularity questions may remain open.

7. Large Deviations, Local Limit Theorems, and Exotic Phenomena

Random walks on free products, under mild conditions (pattern-avoiding driving measures, exponential moments), satisfy large deviation principles for normed distance or word length, with convex rate functions determined as Legendre transforms of limiting logarithmic moment generating functions (Corso, 2020). These rate functions describe the exponential decay of probabilities for deviations from typical escape rates.

Recent work has revealed "exotic" local limit theorems at phase transitions, with new exponents and logarithmic corrections in the asymptotics for return probabilities, especially near spectrally degenerate cases (e.g. free products of $\mathbb{Z}^3 * \mathbb{Z}^5$ or $\mathbb{Z}^3 * \mathbb{Z}^6$ with critical random walk parameters) (Dussaule et al., 2023). This demonstrates incompleteness in prior classifications and signals further complexity in relatively hyperbolic and nonamenable regimes.

8. Range, Capacity, Fractal Boundaries, and Branching Structures

The range (number of distinct visited vertices) and capacity (charge-holding ability of the visited set) have asymptotic linear growth laws and satisfy central limit theorems, with real-analytic dependence on model parameters (Gilch, 2020, Gilch, 2022). Fractal aspects, such as box-counting and Hausdorff dimensions of the trace boundary, exhibit phase transitions and discontinuities associated with the survival parameter in branching random walks, with explicit connections to the radius of convergence of underlying Green's functions (Candellero et al., 2011).

Boundary decompositions, renewal-based fractal geometry, and explicit Perron–Frobenius formulations are all central to the quantitative analysis of random walks in free product settings.

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In summary, random walks on free products present a rich tapestry of probabilistic, geometric, and analytic phenomena. The interplay of group structure, renewal theory, generating functions, sensitivity analyses, fractal boundaries, and asymptotics (CLTs, LDPs, local limit theorems) yields deep and precise results with implications for group theory, random matrix products, noncommutative probability, and dynamical systems (Gilch, 2010, Gilch, 2010, Gilch et al., 2015, Gilch, 20 Oct 2025, Gilch, 2020, Gilch, 2022, Dussaule et al., 2023).