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Tail distributions of cover times of once-reinforced random walks

Xiangyu Huang, Yong Liu, Kainan Xiang

TL;DR

This work analyzes the tail behavior of the edge cover time $C_E$ for the $\delta$-once-reinforced random walk on finite connected graphs, showing an exponential tail with a critical exponent $\alpha_c^1(\delta)$. A lifted directed graph and a generalized Laplace principle are developed to handle the non-Markovian nature of the process, yielding a variational framework in which tail decay rates are characterized by rate-function minimizations. The paper derives variational representations for the critical exponent $\alpha_c(\delta)$ (and $\alpha_c^1(\delta)$) and proves key analytic properties, including monotonicity, continuity, and sharp asymptotics as $\delta \to 0+$ and $\delta \to \infty$, with behavior depending on the underlying graph structure. A stochastic comparison result across reinforcement levels is established, and the methods extend to hitting and subgraph cover times, offering a path toward recurrence questions for reinforced walks. Overall, the work provides a rigorous, variational, large-deviation-based characterization of edge-cover-time tails for a non-Markovian reinforcement model and connects graph structure to the phase transition in exponential integrability.

Abstract

We consider the tail distribution of the edge cover time of a specific non-Markov process, $δ$ once-reinforced random walk, on finite connected graphs, whose transition probability is proportional to weights of edges. Here the weights are $1$ on edges not traversed and $δ\in(0,\infty)$ otherwise. In detail, we show that its tail distribution decays exponentially, and obtain a phase transition of the exponential integrability of the edge cover time with critical exponent $α_c^1(δ)$, which has a variational representation and some interesting analytic properties including $α_c^1(0+)$ reflecting the graph structures.

Tail distributions of cover times of once-reinforced random walks

TL;DR

This work analyzes the tail behavior of the edge cover time for the -once-reinforced random walk on finite connected graphs, showing an exponential tail with a critical exponent . A lifted directed graph and a generalized Laplace principle are developed to handle the non-Markovian nature of the process, yielding a variational framework in which tail decay rates are characterized by rate-function minimizations. The paper derives variational representations for the critical exponent (and ) and proves key analytic properties, including monotonicity, continuity, and sharp asymptotics as and , with behavior depending on the underlying graph structure. A stochastic comparison result across reinforcement levels is established, and the methods extend to hitting and subgraph cover times, offering a path toward recurrence questions for reinforced walks. Overall, the work provides a rigorous, variational, large-deviation-based characterization of edge-cover-time tails for a non-Markovian reinforcement model and connects graph structure to the phase transition in exponential integrability.

Abstract

We consider the tail distribution of the edge cover time of a specific non-Markov process, once-reinforced random walk, on finite connected graphs, whose transition probability is proportional to weights of edges. Here the weights are on edges not traversed and otherwise. In detail, we show that its tail distribution decays exponentially, and obtain a phase transition of the exponential integrability of the edge cover time with critical exponent , which has a variational representation and some interesting analytic properties including reflecting the graph structures.
Paper Structure (10 sections, 12 theorems, 109 equations, 3 figures)

This paper contains 10 sections, 12 theorems, 109 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results and preliminaries
  3. Main results
  4. Lifted directed graph and generalized Laplace principle
  5. Variational representations of critical exponent
  6. Analytic properties and asymptotic behaviors of critical exponent
  7. Proofs of Proposition \ref{['another expression of alpha_c']}
  8. Proof of Lemma \ref{['order of Markov process']}
  9. Proofs of Proposition \ref{['another expression of alpha_c 1']}
  10. Proofs of Proposition \ref{['property of alpha_c 2']}

Key Result

Theorem 1.1

$\mathbb{P}_{x_0}\left(C_E>n\right)$ decays exponentially with the rate $\alpha^1_c(\delta)\in (0,\infty)$, i.e., More precisely, And $\alpha_c^1(\delta)$ is the critical exponent for exponential integrability of $C_E$ in the sense that

Figures (3)

  • Figure 1: This is a sketch map of $\alpha_c^1(\cdot)$ for the ORRW on finite connected graphs. The left picture stands for $\alpha_c^1(\cdot)$ on $3$-vertex connected graph or star-shaped graph. In this picture, $\alpha_c^1(\delta)$ converges to $\infty$ and $0$ as $\delta$ approaches $0$ and $\infty$ respectively. The right picture stands for $\alpha_c^1(\cdot)$ on other graphs. Here, $\alpha_c^1(\delta)$ converges to a positive number as $\delta$ approaches $0$, and converges to $0$ as $\delta$ approaches $\infty$.
  • Figure 2: Under condition $\mathcal{Z}_{\tau_k}|_E\in E_k\ (1\le k\le i-1)$, the probability of the event $\{\tau_{i}-\tau_{i-1}>n_i\}$ has the same order as $p_{i,\{E_j\}}^{n_i}$.
  • Figure 3: (a) and (b) stand for the cases (a) and (b) of ${\mathbf{q}}^\varepsilon$ respectively.

Theorems & Definitions (29)

  • Theorem 1.1: Critical exponent for exponential integrability of $C_E$
  • Theorem 1.2: Analytic property and asymptotic behaviour of $\alpha^1_c(\delta)$
  • Definition 2.1: Transition kernel on $G$
  • Theorem 2.2: Critical exponent for exponential integrability of $\mathcal{T}_{\mathscr{S}_0}$
  • Theorem 2.3: Analytic property and asymptotic behaviour of $\alpha_c(\delta)$
  • Corollary 2.4
  • proof : Proof of Corollary \ref{['stochastic inequality']}
  • Remark 2.5
  • Definition 2.6: Transition kernel on $S$
  • Definition 2.7
  • ...and 19 more