Phase transitions and scaling in an interacting elephant random walk with random amplitudes

Mohamed Abdelkader; Rafik Aguech; Mohamed Abdelkader; Rafik Aguech

doi:10.3934/math.20261240

AIMS Mathematics

2026, Volume 11, Issue 9: 31412-31433. doi: 10.3934/math.20261240

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Research article

Phase transitions and scaling in an interacting elephant random walk with random amplitudes

Mohamed Abdelkader ^,,
Rafik Aguech

Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 04 April 2026 Revised: 22 July 2026 Accepted: 24 August 2026 Published: 23 September 2026
MSC : 60G50, 60F05, 60G42, 60K35, 82C41

We investigated a modified two-elephant random walk in which one walker follows the classical dynamics, while the other recalls past steps of its partner and multiplies them by independent random amplitudes, introducing heterogeneous reinforcement effects. Using martingale techniques, we established a law of large numbers (LLN), a central limit theorem (CLT), and a law of the iterated logarithm (LIL) for the position of the first elephant. A phase transition driven by an effective memory parameter was identified. In both the diffusive and critical regimes, Gaussian fluctuations arise, with a change of scaling at criticality due to logarithmic corrections. In the super-diffusive regime, the walk converges to a non-Gaussian limit at a polynomial rate. Numerical simulations support the theoretical results. Unlike previous interacting elephant walks, we uncovered a non-Gaussian limit in the super-diffusive regime and explicit logarithmic corrections at criticality, highlighting the profound impact of random amplitudes on memory-driven systems.
- elephant random walk,
- non-Markovian processes,
- anomalous diffusion,
- interacting random walks,
- central limit theorem,
- asymptotic normality,
- martingale methods,
- law of large numbers
Citation: Mohamed Abdelkader, Rafik Aguech. Phase transitions and scaling in an interacting elephant random walk with random amplitudes[J]. AIMS Mathematics, 2026, 11(9): 31412-31433. doi: 10.3934/math.20261240

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Abstract

We investigated a modified two-elephant random walk in which one walker follows the classical dynamics, while the other recalls past steps of its partner and multiplies them by independent random amplitudes, introducing heterogeneous reinforcement effects. Using martingale techniques, we established a law of large numbers (LLN), a central limit theorem (CLT), and a law of the iterated logarithm (LIL) for the position of the first elephant. A phase transition driven by an effective memory parameter was identified. In both the diffusive and critical regimes, Gaussian fluctuations arise, with a change of scaling at criticality due to logarithmic corrections. In the super-diffusive regime, the walk converges to a non-Gaussian limit at a polynomial rate. Numerical simulations support the theoretical results. Unlike previous interacting elephant walks, we uncovered a non-Gaussian limit in the super-diffusive regime and explicit logarithmic corrections at criticality, highlighting the profound impact of random amplitudes on memory-driven systems.

References

[1]	G. M. Schütz, S. Trimper, Elephants can always remember: exact long-range memory effects in a non-Markovian random walk, Phys. Rev. E, 70 (2004), 045101(R). https://doi.org/10.1103/PhysRevE.70.045101 doi: 10.1103/PhysRevE.70.045101
[2]	R. Metzler, J. Jeon, A. G. Cherstvy, E. Barkai, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking, Phys. Chem. Chem. Phys., 16 (2014), 24128–24164. https://doi.org/10.1039/c4cp03465a doi: 10.1039/c4cp03465a
[3]	J. Dedecker, X. Fan, H. Hu, F. Merlevède, Rates of convergence in the central limit theorem for the elephant random walk with random step sizes, J. Stat. Phys., 190 (2023), 154. https://doi.org/10.1007/s10955-023-03168-6 doi: 10.1007/s10955-023-03168-6
[4]	R. Aguech, S. Quin, How two elephants can learn from each other, Stochastic Process. Appl., 201 (2026), 105040. https://doi.org/10.1016/j.spa.2026.105040 doi: 10.1016/j.spa.2026.105040
[5]	M. Benaïm, Dynamics of stochastic approximation algorithms, In: Séminaire de Probabilités XXXIII, Berlin: Springer, 1999. https://doi.org/10.1007/BFb0096509
[6]	T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226–1229. https://doi.org/10.1103/PhysRevLett.75.1226 doi: 10.1103/PhysRevLett.75.1226
[7]	R. Pemantle, A survey of random processes with reinforcement, Probab. Surveys, 4 (2007), 1–79. https://doi.org/10.1214/07-PS094 doi: 10.1214/07-PS094
[8]	H. Kesten, V. Sidoravicius, The spread of a rumor or infection in a moving population, Ann. Probab., 33 (2005), 2402–2462. https://doi.org/10.1214/009117905000000413 doi: 10.1214/009117905000000413
[9]	B. Bercu, A martingale approach for the elephant random walk, J. Phys. A Math. Theor., 51 (2018), 015201. https://doi.org/10.1088/1751-8121/aa95a6 doi: 10.1088/1751-8121/aa95a6
[10]	M. Duflo, Random iterative models, Berlin: Springer, 1997.
[11]	P. Hall, C. C. Heyde, Martingale limit theory and its application, New York: Academic Press, 1980. https://doi.org/10.1016/C2013-0-10818-5
[12]	W. F. Stout, Almost sure convergence, New York: Academic Press, 1974.

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