Transition probabilities of step-reinforced random walks

Abstract: The step-reinforced random walk (SRRW), where each step may replicate a randomly chosen past step, exhibits complex dependencies on the history. This paper introduces a generalized SRRW on groups, incorporating arbitrary transformations of past steps, which unifies several existing models in the literature. We develop a unified framework for establishing upper bounds on its transition probabilities for any reinforcement parameter $α<1$, linking the decay rate directly to the geometry of the underlying group. We prove that on Euclidean space, the walk is transient in all dimensions $d \geq 3$ for any $α<1$. On finitely generated groups, we derive the upper bounds using the isoperimetric profile of the Cayley graph, which in particular resolves an open problem regarding the exponential decay of the elephant random walk on Cayley trees.

• The paper presents a unified SRRW model that incorporates arbitrary reinforcement and transformation, enabling broad transition probability analysis.
• Transition probability bounds are derived using evolving set methods and recursive tree percolation, demonstrating polynomial or exponential decay based on group structure.
• The work proves transience in dimensions three and higher and resolves open problems by leveraging advanced Markov chain and isoperimetric techniques.

Transition Probabilities of Step-Reinforced Random Walks

Introduction and Generalized Framework

Step-reinforced random walks (SRRWs) are non-Markovian stochastic processes wherein, at each step, the walker either repeats a randomly chosen previous increment or takes a fresh independent step. The work "Transition probabilities of step-reinforced random walks" (2604.07227) presents a unified and generalized SRRW model on groups, parameterized by arbitrary step reinforcement and transformations, including as special cases various previously studied models such as the elephant random walk (ERW), echoed step processes, and random walks with balanced or counterbalanced reinforcement.

The core generalization allows the reinforced choice not only to copy but also to transform a randomly selected previous step via (possibly random) group automorphisms. The model thus synthesizes additive, multiplicative, and invertible transformations, providing a broad unifying structure that captures both historical dependency and structural symmetries or asymmetries of the underlying state space.

Main Results: Transition Probability Bounds and Transience

Transition Probability Decay: Euclidean and Group Cases

For SRRWs on $\mathbb{R}^d$ with genuine $d$-dimensional step distributions and reinforcement parameter $\alpha < 1$, the authors establish a universal upper bound for the $r$-ball return probability:

$P(\|S_n\| < r) \leq C n^{-d/2}$

for a constant $C$ depending on $r$, $\alpha$, and the step law $\mu$. Crucially, this holds for any $\alpha < 1$ and arbitrary transformations, resolving previous constraints on higher-order moments in prior work and confirming transience in all dimensions $d$0. This aligns the behavior of generalized SRRWs with standard random walks in high dimension, despite their non-Markovian dependency.

Cayley Graphs: Isoperimetric Profile and Exponential Decay

For finitely generated (possibly infinite) groups, transition probability upper bounds are derived in terms of the isoperimetric (bottleneck) profile $d$1 of the Cayley graph. Specifically, when the step distribution is supported on a finite symmetric generating set and the graph is nonamenable, it is shown that

$d$2

for some $d$3, demonstrating exponential decay of return probabilities. This settles an open problem for ERWs on Cayley trees, affirming that the $d$4-step return probability undergoes exponential decay for all memory parameter $d$5 and degree $d$6.

For groups of polynomial or exponential growth, the decay is polynomial or stretched-exponential, respectively, consistent with standard isoperimetric inequalities for Markov chains and lazy random walks, but here established non-Markovianly via evolving sets and recursive tree representations.

Technical Innovations

Recursive Tree Percolation Representation

A central technical device is the construction of a random forest (specifically, a random recursive tree under Bernoulli percolation) that encapsulates the dependence structure of the SRRW path. Each vertex corresponds to a time index, and the rules for percolation encode whether each step is a 'fresh' step or a reinforced (possibly transformed) copy. Analysis of the cluster sizes (notably the isolated vertices) reduces the multistep, non-Markovian dependency to conditional Markovian evolution across a collection of random times.

Evolving Set and Doob Transform Methods

The paper adapts sophisticated Markov chain mixing tools (evolving set process à la Morris and Peres, Doob transforms, isoperimetric functional analysis) to the inhomogeneous and history-dependent SRRW setting, upper bounding the $d$7-norms of transition kernels via bottleneck ratios and relating the impact of reinforcements to the minimum number of fresh steps in a trajectory.

Explicit Estimates and Elephant Polynomials

For the two-point case and cycle groups, explicit asymptotics for return probabilities are given in terms of elephant polynomials $d$8, with exponential decay in the nondegenerate uniqueness regime. Upper and lower bounds for the leading coefficients in these polynomials are provided, yielding precise rates such as:

$d$9

for the uniform SRRW on $\alpha < 1$0, with $\alpha < 1$1.

Implications and Future Directions

The unified framework allows the extension of mixing, transience, and spectral methods from Markov chains to large classes of reinforced, history-dependent walks. This facilitates:

• Generalization of mixing time and convergence results to reinforced stochastic processes beyond the Markov context,
• Precise characterization of phase transitions between recurrence/transience as a function of reinforcement and group geometry,
• Transfer of functional limit theorems and fluctuation analysis from urn models and recursive trees to non-Markovian random walks,
• Resolution of open problems concerning exponentially fast escape (or localization) on nonamenable graphs, with explicit constants.

Potential future work includes quantitative refinements of CLT and moderate deviation principles for SRRWs in critical/transient regimes [4646949], universality conjectures for scaling limits [bertenghi2024universal], and extension of the approach to other classes of reinforced processes (e.g., vertex/edge reinforcement, interacting reinforced systems).

Conclusion

This work establishes a rigorous probabilistic and analytic machinery for step-reinforced random walks with arbitrary group-valued transformations, connecting probabilistic, combinatorial, and spectral tools. Key results include sharp upper bounds on transition probabilities, demonstration of transience and mixing properties across wide generality, and explicit asymptotic rates. These findings resolve several standing questions in the field of self-interacting random walks, and provide a flexible methodology applicable to a broad spectrum of reinforced stochastic models (2604.07227).