Two-Elephant Walking Model Dynamics

• The Two-Elephant Walking Model is a framework for coupled, memory-dependent random walks where each elephant’s move is influenced by both its own and its partner’s past actions.
• It employs reinforcement parameters and spectral techniques to classify regimes into diffusive, critical, and superdiffusive behaviors based on memory effects.
• Extensions of the model include graph-based coupling, exclusion dynamics, and connections to random recursive trees, illustrating broad applications in stochastic processes.

The Two-Elephant Walking Model describes a class of coupled, memory-dependent random walks in which two agents ("elephants") on a discrete time axis update their positions with rules that reflect not just their own past, but also the past of their partner. This model encapsulates both the non-Markovian reinforcement typical of the elephant random walk (ERW) and inter-agent interaction, producing new regimes and phenomena in stochastic processes with memory.

1. Formal Specification and Reinforcement Structure

The two-elephant walking model tracks positions $S_n^{(1)}$ and $S_n^{(2)}$ at time $n$ for elephants $1$ and $2$. Each elephant's step is determined by the past steps of its partner, subject to memory parameters:

• Define $p_1$ and $p_2$ as the probabilities that elephant $1$ and $2$, respectively, repeat their partner’s previous step when choosing that partner's history.
• Set $\alpha_1 = 2p_1 - 1$, $\alpha_2 = 2p_2 - 1$ as reinforcement parameters.

At each time step, the dynamics proceed recursively. The canonical form is: $$\text{For each } n,\quad S_{n+1}^{(1)} = S_n^{(1)} + \xi_{n+1}^{(1)} \ S_{n+1}^{(2)} = S_n^{(2)} + \xi_{n+1}^{(2)}$$ where the increments $\xi_{n+1}^{(i)}$ ($i=1,2$) are random variables, whose distributions depend on the full history of the partner's process. For instance, $\xi_{n+1}^{(1)}$ is generated by selecting a time $k\leq n$ uniformly at random and then, with probability $p_1$, repeating the step taken by $S_k^{(2)}$, and with probability $1-p_1$, inverting it.

By mapping to a stochastic approximation recursion and decomposing using appropriate linear transformations, the coupled system is represented in coordinates: $$x_n = \frac{S_n^{(1)} - r_\alpha S_n^{(2)}}{n} \ y_n = \frac{S_n^{(1)} + r_\alpha S_n^{(2)}}{n}$$ with $r_\alpha = \sqrt{\alpha_1/\alpha_2}$ when $\alpha_1\alpha_2 > 0$, $r_\alpha = i\sqrt{-\alpha_1/\alpha_2}$ when $\alpha_1\alpha_2 < 0$.

2. Regime Classification via Reinforcement Parameters

The long-term and fluctuation behavior is determined by the regime of the reinforcement parameters:

• Diffusive Regime: $|\lambda_\alpha| < 1/2$, with $\lambda_\alpha$ the principal eigenvalue of the associated drift matrix (constructed from $\alpha_1$ and $\alpha_2$). Under this regime, the system exhibits classical law of large numbers and central limit theorem scaling:

$$\frac{S_n^{(1)}}{\sqrt{n}} \to \mathcal{N}\Big(0,\ \sigma^2(\alpha_1, \alpha_2)\Big)$$

The covariance $\sigma^2$ is explicitly characterized in terms of the memory parameters.

• Critical Regime: $|\lambda_\alpha| = 1/2$. Scaling is $n^{-1/2}(\log n)^{-1/2}$ and fluctuations also converge to a Gaussian, but logarithmic corrections appear.
• Superdiffusive Regime: $|\lambda_\alpha| > 1/2$. After suitable normalization by $n^{|\lambda_\alpha|}$, the process exhibits almost sure convergence to a non-degenerate (non-Gaussian) random variable or vector, determined by martingale techniques and recursive tree connections.

3. Connection to Random Recursive Trees and Martingale Methods

A distinctive aspect is the connection to random recursive trees (RRT):

• Each time step in the walk relates to a node in a RRT.
• The out-degree $d_n(k)$ of vertex $k$ in a RRT up to time $n$ enters into alternative formulations for the position: $S_n^{(1)}$ can be written as a weighted sum involving these degrees.
• Known asymptotic properties of $d_n(k)$ (e.g., $d_n^* / \log_2(n) \to 1$ in probability) feed into precise analysis of the walk's behavior.

Analysis is based on representing linear combinations of normalized positions as weighted sums of martingale differences. Quadratic variations and their limits enable exact determination of law-of-large-numbers and central limit theorem typologies.

4. Generalizations: Coupled and Block-Structured Memory

More abstract formulations allow for richer coupling:

• Each elephant's update may depend on convex mixtures of their own and their partner's histories, as encoded by coefficients $\gamma_k^i$.
• For the "cow-and-ox" subclass (where the memory structure is asymmetric), the first-order dynamics of the "ox" are governed by

$$\langle x_{t+1}^2 \rangle = \left(\frac{\gamma \alpha_1^2}{t}\right) \langle x_t^1 \rangle + \left[1 + \frac{(1-\gamma)\alpha_2^2}{t}\right] \langle x_t^2 \rangle$$

• New regimes can arise, such as "following" (the ox’s displacement tracks the cow’s) or "antagonistic" (motion in oppositional directions).

The Fokker-Planck formalism yields explicit drift expressions for joint positions, revealing additional superdiffusive regimes not seen in the single-walker case.

5. Interacting Models and Exclusion Dynamics

When two elephants interact on a discrete lattice with exclusion (each site at most singly occupied), as in reinforced exclusion processes:

• The model admits ballistic, sub-ballistic, or condensed phases, with phase transitions controlled by the memory parameters and initial configurations.
• For two particles, mutual exclusion introduces nontrivial two-body correlations, which can analogize "pair formation" or transient clustering found in the many-particle case.

Mean-field and local mean-field analyses yield explicit predictions for particle current and density, with self-consistency equations derived for the jump probabilities and cluster sizes.

6. Extensions: Graph-Based and Generalized Coupling

Graph-encoded interaction models generalize to multiple elephants on a network:

• Each elephant $i$ references its in-neighbors (defined by a graph $\mathcal{G}$), and memory-driven reinforcement becomes entry-wise structured by a "memory matrix" $B$.
• In the two-elephant case, $$B = \begin{pmatrix} 0 & 2p-1 \ 2p-1 & 0 \end{pmatrix}$$, and the system is analyzed via stochastic approximation, leading to diffusive, critical, or superdiffusive scaling depending on $p$.

Advanced results include strong invariance principles and central limit theorems, with error rates and fluctuation magnitudes governed by spectral properties of $B$.

7. Analytical Consequences, Regimes, and Phase Transitions

The two-elephant model illustrates core phenomena from non-Markovian reinforcement and mutual memory:

• Memory can induce synchronization, locking, or even persistent separation in the walks.
• Sensitivity to initial conditions becomes marked if the reinforcement is strong or multiple-history majority rules are in effect (generalized ERW extracting $k$ prior steps).
• In majority-of-$k$ extraction with $k>2$ and $p$ above a critical threshold, symmetry breaking occurs: the walk converges with high probability to one of two dynamically stable attractors, reflecting the initial asymmetry—a lock-in analogous to path dependence in learning models.

Large deviation entropy in these multiple-extraction models can become flat between attractors, indicating sublinear entropic penalty—i.e., almost deterministic choice of trajectory after coordination emerges.

8. Summary Table: Regimes and Their Features

Memory Parameter(s)	Asymptotic Scaling	Asymptotic Law
$|\lambda_\alpha| < 1/2$	$n^{-1/2}$	Gaussian diffusive CLT
$|\lambda_\alpha| = 1/2$	$[n\log n]^{-1/2}$	Log-corrected CLT
$|\lambda_\alpha| > 1/2$	$n^{-|\lambda_\alpha|}$	Power law, non-Gaussian
$k>2, p>p_c$ (majority)	Deterministic attractors, flat entropy	Initial-condition lock-in

Interpretation of these regimes is entwined with the eigenstructure of the coupling/memory matrix and the combinatorics of memory extraction rules.

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The two-elephant walking model exposes the rich interplay of reinforcement, memory, and interaction in non-Markovian stochastic processes. Its mathematical structure is governed by spectral and combinatorial properties derived from coupling matrices, recursive tree representations, and nonlinear reinforcement rules, leading to foundational anomalous regimes, synchronization phenomena, and sensitivity to early histories.