Generalized step-reinforced random walks on groups admit transition probability upper bounds determined by the isoperimetric profile of the Cayley graph, proving transience in Euclidean space for dimensions d≥3 and any reinforcement α<1, and resolving exponential decay for the elephant random walk.
On the multi-dimensional elephant random walk.J
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Step-reinforced random walks on finite groups converge exponentially to uniform; on cycles mixing time jumps from logarithmic to polynomial at alpha=1/2, while on hypercubes reinforcement slows mixing with cutoff at d log d over F(alpha)(1-alpha).
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Transition probabilities of step-reinforced random walks
Generalized step-reinforced random walks on groups admit transition probability upper bounds determined by the isoperimetric profile of the Cayley graph, proving transience in Euclidean space for dimensions d≥3 and any reinforcement α<1, and resolving exponential decay for the elephant random walk.
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Mixing times of step-reinforced random walks
Step-reinforced random walks on finite groups converge exponentially to uniform; on cycles mixing time jumps from logarithmic to polynomial at alpha=1/2, while on hypercubes reinforcement slows mixing with cutoff at d log d over F(alpha)(1-alpha).