Anomalous Diffusion and Emergent Universality in Coupled Memory-Driven Systems

Understanding how simple local interactions give rise to emergent exploration patterns is a fundamental question in statistical physics. We introduce a minimal model of two coupled agents that avoid retracing their own paths while being attracted to the trails left by one another. This system is inspired by, but not limited to, pheromone-guided insect navigation. The coupling of self-avoidance and attraction generates rich emergent behavior, including distinct anomalous diffusion regimes, non-Gaussian position distributions, and compressed exponential encounter statistics. Most notably, we identify new universality classes for coupled random walks, characterized by unique scaling laws and distributional properties that, to our knowledge, have not been previously reported. These findings advance the theoretical understanding of coupled stochastic processes with memory and interaction feedback, providing a framework for exploring transport phenomena in a broad range of multi-agent systems beyond biological contexts.

💡 Research Summary

This paper presents a groundbreaking investigation into the emergent collective dynamics and novel universality classes arising from coupled stochastic processes with memory and interaction feedback. Inspired by pheromone-mediated navigation in insects, the authors introduce a minimalist model of two interacting agents moving on a lattice. The core innovation lies in the agents’ behavioral rule: each agent is repelled by its own past trail (self-avoidance with strength β) while simultaneously being attracted to the trail left by the other agent (mutual attraction with strength β’). This creates a dynamic, shared environment where the agents’ movements continuously modify the landscape that guides their future decisions.

The model is rigorously defined mathematically. The transition probability for an agent to move to a neighboring site depends exponentially on the difference between the attraction to the other agent’s “debris” count at that site and the repulsion from its own. The study focuses on key observables: the mean-squared displacement (MSD) of a single agent, the mean-squared distance between the two agents, and the probability distributions of their positions and encounter statistics.

Through extensive Monte Carlo simulations in one and two dimensions, the authors uncover rich phase diagrams dictated by the ratio β’/β. In one dimension, three distinct universality classes are identified: 1) A superdiffusive regime (β’ ≤ β) where the MSD scales as t^(4/3), matching the exponent of the true self-avoiding walk but arising from a different, interactive mechanism. 2) A pseudo-normal/subdiffusive regime (β’ > B(β)) where the diffusion exponent α approaches 1 from below (α → 1-), accompanied by non-Gaussian position distributions. 3) A distinct subdiffusive regime (β=0, β’ large) with clear α = 1/2 scaling. The relative distance between agents shows analogous anomalous diffusion. In two dimensions, the scaling behavior incorporates logarithmic corrections, e.g., R²_t ~ t (ln t)^(1/2) for β ≤ β'.

Crucially, the universality classes extend beyond scaling exponents. The shapes of the position probability distributions P(x,t)—such as a double-peak structure for certain parameters—and the statistics of agent encounters (number and duration) also exhibit distinct, universal forms characteristic of each dynamical phase. This demonstrates that the microscopic feedback mechanism profoundly shapes the macroscopic statistical properties.

The work significantly advances the theoretical understanding of coupled non-Markovian processes. It provides a powerful framework for analyzing transport and search phenomena in a wide array of systems beyond biology, including active colloids, growing neural networks, autonomous robotic teams, and algorithms for optimization or resource exploration, where agents interact through a dynamically modified common environment.