Stochastic Dynamics of Reinforced Random Walks

Reinforced random walks constitute a class of self-interacting processes in which past visits enhance future probabilities of traversal. Over the past decades, two main variants have emerged: edge-reinforced random walks, where edges accumulate weight with each crossing, and vertex-reinforced jump processes, where vertices gain attractiveness proportional to accumulated local time. These models exhibit rich phase behaviour, including transitions between recurrent regimes—where the walker returns indefinitely to the origin—and transient regimes—where escape to infinity occurs with unit probability. In low dimensions or on tree-like structures, strong reinforcement often induces trapping and localisation, whereas in higher dimensions a delicate balance between exploration and reinforcement gives rise to intermediate regimes. Recent theoretical advances have leveraged supersymmetric representations, branching random walk techniques and probabilistic identities to characterise critical thresholds, quantify return-time divergences and uncover novel intermediate scales. Applications span statistical physics, where reinforced walks serve as toy models for glassy dynamics, to network science and ecology, modelling preferential attachment and trail formation in animal foraging. Ongoing research seeks to unify discrete-time and continuous-time formulations, to extend characterisations to dynamic and directed environments, and to establish universal scaling laws governing reinforced interactions on complex geometries.

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Stochastic Dynamics of Reinforced Random Walks publication trend

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Technical terms

Edge-reinforced random walk: A discrete-time process in which each traversal increases the weight of that edge, biasing future moves.

Vertex-reinforced jump process (VRJP): A continuous-time walk that preferentially jumps to vertices in proportion to accumulated local time at each vertex.

Local time: The cumulative amount of time or number of visits a walk has spent at a given site or edge.

Recurrence and transience: Recurrence refers to almost-sure infinite returns to a starting point; transience denotes eventual escape with probability one.

References

1. H2|2-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase. Communications in Mathematical Physics (2024).
2. Inverting Ray-Knight identities on trees. Electronic Journal of Probability (2024).
3. The Directed Edge Reinforced Random Walk: The Ant Mill Phenomenon. Journal of Statistical Physics (2022).