Mixing times of step-reinforced random walks

Abstract: We study the mixing time of a non-Markovian process -- step-reinforced random walk -- on a finite group $G$. This process differs from a classical random walk on $G$ in that at each integer time, with probability $α$ the next step is chosen uniformly from the previous steps of the walk. We prove that the distribution of the walk converges to the uniform distribution exponentially fast if the walk is irreducible and aperiodic. Some quantitative bounds are provided when the non-reinforced chain is lazy, or when the step distribution is symmetric or a class function. We also show that the reinforced simple random walk on cycles undergoes a phase transition at $α=1/2$ and the reinforcement can speed up mixing for $α> 1/2$.

• The paper establishes exponential convergence bounds for SRRW, uncovering phase transitions in mixing behavior based on the reinforcement parameter.
• It innovatively connects SRRW to dynamic percolation on random recursive trees by employing concentration inequalities and evolving set techniques.
• Sharp mixing time bounds are derived from isoperimetric and spectral properties, demonstrating diffusive, subdiffusive, and superdiffusive regimes.

Mixing Times of Step-Reinforced Random Walks on Finite Groups

Introduction and Model Specification

The paper develops a rigorous analysis of mixing times for step-reinforced random walks (SRRW) on finite groups. SRRW are natural non-Markovian generalizations of classical random walks where, at each time step, the walker either takes a fresh step sampled independently from a fixed distribution $\mu$ or, with probability $\alpha$, repeats one of its previous steps (chosen uniformly at random). This "step reinforcement" interpolates between the usual i.i.d. random walk ($\alpha = 0$) and a deterministic walk with maximal memory ($\alpha = 1$). The SRRW is parametrized by the reinforcement parameter $\alpha \in [0,1)$, step distribution $\mu$ (which induces the "base" Markov chain $P_\mu(x,y) = \mu(x^{-1}y)$), and its underlying group structure.

Previous work has extensively addressed SRRW and related processes on $\mathbb{Z}^d$ and real spaces, classifying diffusive and superdiffusive regimes and limit theorems. The present paper initiates a systematic analysis of SRRW on general finite groups, with focus on their convergence (in total variation) to the uniform distribution and quantification of the mixing time, $t_\mathrm{mix}^{(\alpha)}(\varepsilon)$.

Main Technical Tools

A core innovation is the connection of SRRW to dynamic percolation on random recursive trees, generalizing prior links recognized in the case of the elephant random walk. The SRRW can be re-expressed as a mixture of time-inhomogeneous Markov chains, the (random) emphemeral steps corresponding to isolated vertices in a percolated tree. The main analytical contribution is a fine-grained enumeration and control of these "free" steps using concentration inequalities, martingale techniques, and evolving set methodologies.

The analysis develops several structural results:

• For irreducible and aperiodic $\mu$, the SRRW's mixing time is finite and obeys explicit exponential convergence bounds, with strong dependence on both group structure and $\alpha$0.
• The mixing time can be sharply characterized in several regimes, showing both speed-up and slowdown as $\alpha$1 varies and as $\alpha$2 varies.

Exponential Convergence and Phase Transition

A principal result establishes that for any irreducible and aperiodic SRRW on a finite group, the total variation distance to equilibrium exhibits exponential decay: $\alpha$3 for some $\alpha$4 independent of $\alpha$5 (and, under mild structural assumptions, independent of $\alpha$6). In particular, the mixing time scales as $\alpha$7 for generic step distributions.

However, the study uncovers a phase transition in mixing behavior for SRRW on finite cyclic groups $\alpha$8, particularly on odd cycles. For SRRW with symmetric, irreducible $\alpha$9 (e.g., nearest-neighbor steps on the cycle),

• If $\alpha = 0$0, the mixing is diffusive, $\alpha = 0$1.
• If $\alpha = 0$2, sub-diffusive correction appears, $\alpha = 0$3.
• If $\alpha = 0$4, superdiffusive behavior emerges with $\alpha = 0$5, reflecting long-range propagation from reinforcement.

The regime $\alpha = 0$6 thus witnesses a strict speedup compared to the Markov ($\alpha = 0$7) case, with the walk rapidly decorrelating due to "macroscopic" step repetitions. This is in sharp contrast to the behavior on structures such as the hypercube, where reinforcement slows mixing.

Mixing Time Bounds: Group Structure and Step Distribution

The paper provides sharp upper and lower bounds on $\alpha = 0$8 in terms of both isoperimetric and spectral properties of $\alpha = 0$9:

• Isoperimetric bounds: When $\alpha = 1$0 is nontrivial, the total variation and uniform mixing times are controlled in terms of the isoperimetric profile $\alpha = 1$1 via evolving sets,

$\alpha = 1$2

• Spectral bounds: When $\alpha = 1$3 is symmetric, the spectral gap $\alpha = 1$4 gives,

$\alpha = 1$5

The technique also identifies necessary and sufficient algebraic conditions on the support of $\alpha = 1$6 for the mixing time to display "homogenized" dependence on $\alpha = 1$7, i.e., existence of global exponential contraction regardless of $\alpha = 1$8.

Model Connections and Special Cases

The authors unify SRRW and a broad range of reinforced Markov processes. The connection to random recursive trees, where the backbone of the process is built via reinforced steps (edges retained under percolation), provides a powerful coupling to classical Markov chains and enables application of concentration inequalities and spectral techniques.

Connections are also drawn to:

• Pólya urn models, which underlie the long memory present in reinforcement.
• Transitional behavior in related reinforced processes (e.g., VRRW and once/edge-reinforced walks), where structural phase transitions and localization phenomena are prominent.

Implications and Theoretical Perspectives

The analysis demonstrates that step reinforcement can induce qualitatively new mixing regimes in finite, symmetric structures, with both substantial speedup and slowdown possible depending on group topology and walk parameters. In some cases (cycles), memory increases ergodicity, while in others (hypercubes, large Abelian groups), the increased memory hinders mixing.

This dichotomy raises fundamental questions about the role of reinforcement and memory in non-Markovian stochastic processes on networks. The present framework offers:

• Explicit control of non-Markovian randomness via time-inhomogeneous Markov chain representations.
• Quantitative tools for analyzing reinforced processes beyond classical diffusive paradigms.

Further implications include connections to reinforced learning models in adaptive systems, synchronization phenomena in interacting reinforced urns, and algorithmic randomization schemes with memory.

Conclusion

This work provides a comprehensive, rigorous framework for understanding the mixing times of step-reinforced random walks on finite groups (2604.07207). Through a blend of combinatorial, spectral, and probabilistic approaches, it identifies phase transitions, sharp mixing time bounds, and the structural mechanisms governing the ergodicity of non-Markovian random walks under reinforcement. The techniques and results presented set the stage for further inquiries into reinforced processes in both finite and infinite-state environments, with potential applications ranging from stochastic algorithms to statistical physics and reinforced learning.