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Phase Transitions for Elephant Random Walks with Two memory Channels

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arxiv 2509.10225 v1 pith:F5LL4RBK submitted 2025-09-12 math.PR cond-mat.stat-mech

Krishanu Maulik , Parthanil Roy , Tamojit Sadhukhan This is my paper · ORCID
classification math.PRcond-mat.stat-mech
keywords memoryrandomphasestepsuperdiffusivetransitionwalkwalks
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Elephant random walk, introduced to study the effect of memory on random walks, is a novel type of walk that incorporates the information of one randomly chosen past step to determine the future step. However, memory of a process can be multifaceted and can arise due to interactions of more than one underlying phenomena. To model this, random walks with multiple memory channels were introduced in the statistical physics literature by Saha (2022) - here the information on a bunch of independently chosen past steps is needed to decide the future step. With the help of variance heuristics, this work analyzed the two-channel case and predicted a double phase transition: from diffusive to superdiffusive and from superdiffusive to ballistic regimes. We prove these conjectures rigorously (with some corrections), discover a mildly superdiffusive regime at one of the conjectured transition boundaries, and observe a new second-order phase transition. We also carry out a detailed investigation of the asymptotic behavior of the walk at different regimes.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Elephant-Reinforced Galves--L\"ocherbach Networks

    math.PR 2026-08 conditional novelty 6.0 of 10

    A Galves-Loecherbach spiking network with bounded elephant-style synaptic reinforcement is shown to be non-explosive, conditionally Wasserstein-contractive, and to admit a conditional replica mean-field equation under...

  2. Tampered Memory Elephant Random Walk on One-Dimensional Integer Lattice

    math.PR 2026-07 conditional novelty 6.0 of 10

    For tampered-memory elephant random walks, lim |D^c_n|/n = 1/2 is the sharp threshold separating persistence of diffusive/critical/superdiffusive phases from purely diffusive O(√n) behavior.

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