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Large deviation principle for empirical measures of once-reinforced random walks on finite graphs

Xiangyu Huang, Yong Liu, Kainan Xiang

TL;DR

This work addresses the large deviation behavior of empirical measures for δ-ORRWs on finite connected graphs, a non-Markovian process due to edge reinforcement. It overcomes non-Markovianity by lifting the process to a directed graph S and formulating a restricted Laplace functional that yields a generalized Laplace principle, enabling a variational rate function I_δ. A central finding is a δ-dependent phase transition at δ_c = 1: I_δ is constant for δ ≤ 1 and strictly decreasing for δ > 1, with non-differentiability at δ = 1 and starting-point sensitivity for δ > 1. The paper also provides explicit analyses for trees, star-shaped graphs, and paths, and outlines extensions to other statistics via the generalized Laplace framework and trajectory decomposition into renewal blocks, laying groundwork for broader applications in edge measures and cover times.

Abstract

$δ$ once-reinforced random walks ($δ$-ORRWs) are a type of self-interacting random walks defined on connected graphs with reinforcement factor $δ>0$, moving to neighbours for each step. The transition probability is proposional to weights of edges, where the weights are $1$ on edges not traversed and $δ$ otherwise. In this paper, we show a general version of large deviation principle for empirical measures of $δ$-ORRWs on finite connected graphs by a modified weak convergence approach. We also obtain a phase transition of the rate function of large deviation principle with critical point $δ_c=1$.

Large deviation principle for empirical measures of once-reinforced random walks on finite graphs

TL;DR

This work addresses the large deviation behavior of empirical measures for δ-ORRWs on finite connected graphs, a non-Markovian process due to edge reinforcement. It overcomes non-Markovianity by lifting the process to a directed graph S and formulating a restricted Laplace functional that yields a generalized Laplace principle, enabling a variational rate function I_δ. A central finding is a δ-dependent phase transition at δ_c = 1: I_δ is constant for δ ≤ 1 and strictly decreasing for δ > 1, with non-differentiability at δ = 1 and starting-point sensitivity for δ > 1. The paper also provides explicit analyses for trees, star-shaped graphs, and paths, and outlines extensions to other statistics via the generalized Laplace framework and trajectory decomposition into renewal blocks, laying groundwork for broader applications in edge measures and cover times.

Abstract

once-reinforced random walks (-ORRWs) are a type of self-interacting random walks defined on connected graphs with reinforcement factor , moving to neighbours for each step. The transition probability is proposional to weights of edges, where the weights are on edges not traversed and otherwise. In this paper, we show a general version of large deviation principle for empirical measures of -ORRWs on finite connected graphs by a modified weak convergence approach. We also obtain a phase transition of the rate function of large deviation principle with critical point .
Paper Structure (31 sections, 29 theorems, 238 equations, 13 figures, 2 tables)

This paper contains 31 sections, 29 theorems, 238 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Results on the LDP of ORRWs
  3. Related works on the LDP
  4. Outline of the paper
  5. Main results and preliminaries
  6. Main results
  7. Lifted directed graph and generalized Laplace principle
  8. Brief introduction to weak convergence approach to homogeneous Markov chains
  9. Modified weak convergence approach
  10. Notations
  11. LDP and properties of the rate function
  12. The upper bound of Theorem \ref{['estimate for rate']}
  13. The lower bound of Theorem \ref{['estimate for rate']}
  14. Proof of Theorem \ref{['I_delta']}
  15. Properties of $\Lambda_\delta$ and $I_\delta$
  16. ...and 16 more sections

Key Result

Theorem 1.1

For $\vert V\vert\geq 3$ (equivalently $\vert E\vert\geq 2$), the empirical measure process $(L^n)_{n\geq 1}$ of the $\delta$-ORRW satisfies an LDP with a good rate function $I_{\delta}$. Here the rate function $I_{\delta}$ is continuous and decreasing in $\delta\in [1,\infty)$, and is constant in $

Figures (13)

  • Figure 1: This figure shows a case of $\{E_k\}_k\in\mathscr{E}$ for the graph with vertex set $\{v_i:1\le i\le 4\}$, where $v_i\sim v_{i+1}$ for $1\le i\le 3$ and $v_2\sim v_4$.
  • Figure 2: This figure gives an example of a finite connected graph and the corresponding directed graph.
  • Figure 3: This figure shows the weights in Figure \ref{['state space']} if only edge $e_1$ has been traversed by $X$. The transition probability of $(\mathcal{Z}_n)_{n\geq 0}$ is $p_{\{e_1\}}$, which is induced by weights of vertices.
  • Figure 4: The process $\overline{\mathcal{Z}}_j^n|_1$ stays in $E_k$ during time $\overline{\tau}_{k}^n$ and $\overline{\tau}_{k+1}^n-1$
  • Figure 5: When $\left(\overline{\mathcal{Z}}_\cdot^n\right)^+$ stays in the solid lines in (a), it is a Markov process. Then $\left(\overline{\mathcal{Z}}_\cdot^n\right)^+$ traverses the first dashed line from the left side in (a) at the renewal time, and stays in the solid lines in (b) for a period of time, during which the process is also Markovian.
  • ...and 8 more figures

Theorems & Definitions (67)

  • Theorem 1.1
  • Definition 2.1: Transition kernel on $G$
  • Theorem 2.2: LDP for $(L^n)_{n\geq 1}$
  • Theorem 2.3: Analytic property of $I_\delta$
  • Remark 2.4
  • Definition 2.5: Transition kernel on $S$
  • Definition 2.6
  • Theorem 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 57 more