We studied a model of two interacting elephant random walks through interacting generalized Pólya urns. Each elephant evolves according to both its own memory and the past behavior of the other through interaction parameters $ \alpha $ and $ \beta $. Using spectral methods, we established almost sure convergence and identified a phase transition parameter $ \lambda^\ast $ separating diffusive, critical, and superdiffusive regimes. Depending on the value of $ \Re(\lambda^\ast) $, we derived central limit theorems and anomalous almost sure scaling limits. Several special interaction structures were analyzed, including the fully cross-dependent case, where the two walks synchronize asymptotically and exhibit stable limiting laws. We also discussed extensions with random interaction coefficients and established a large deviation principle for the empirical proportion of positive steps in the diffusive regime.
Citation: Mohamed Abdelkader, Rafik Aguech. Two elephant random walks interacted[J]. AIMS Mathematics, 2026, 11(8): 26097-26119. doi: 10.3934/math.20261045
We studied a model of two interacting elephant random walks through interacting generalized Pólya urns. Each elephant evolves according to both its own memory and the past behavior of the other through interaction parameters $ \alpha $ and $ \beta $. Using spectral methods, we established almost sure convergence and identified a phase transition parameter $ \lambda^\ast $ separating diffusive, critical, and superdiffusive regimes. Depending on the value of $ \Re(\lambda^\ast) $, we derived central limit theorems and anomalous almost sure scaling limits. Several special interaction structures were analyzed, including the fully cross-dependent case, where the two walks synchronize asymptotically and exhibit stable limiting laws. We also discussed extensions with random interaction coefficients and established a large deviation principle for the empirical proportion of positive steps in the diffusive regime.
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Mohamed Abdelkader, Rafik Aguech. Two elephant random walks interacted[J]. AIMS Mathematics, 2026, 11(8): 26097-26119. doi: 10.3934/math.20261045
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