Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of rescaled "true" self-avoiding walks to the Tóth-Werner "true" self-repelling motion

Elena Kosygina, Jonathon Peterson

TL;DR

The paper proves that the rescaled TSAW on $\mathbb{Z}$ converges to the TSRM of Tóth and Werner, establishing a functional limit theorem via a joint generalized Ray-Knight theorem for the rescaled edge local times and their merge/absorption points. The core method inverts the GRKT to deduce convergence of the entire TSAW+local-time process, not merely finite-dimensional distributions, and relies on forward/backward Ray-Knight curves and Brownian-web-type structure to control coalescence events. This yields a robust framework for proving scaling limits of self-interacting walks by leveraging joint Ray-Knight representations and urn-process couplings. The results illuminate a general pathway for deriving process-level limits for self-interacting random walks whenever a GRKT is available, with potential applicability to generalized TSAW models and related edge-local-time processes.

Abstract

We prove that the rescaled ``true'' self-avoiding walk $(n^{-2/3}X_{\lfloor nt \rfloor})_{t\in\mathbb{R}_+}$ converges weakly as $n$ goes to infinity to the ``true'' self-repelling motion constructed by Tóth and Werner. The proof features a joint generalized Ray-Knight theorem for the rescaled local times processes and their merge and absorption points as the main tool for showing both the tightness and convergence of the finite dimensional distributions. Thus, our result can be seen as an example of establishing a functional limit theorem for a family of processes by inverting the joint generalized Ray-Knight theorem.

Convergence of rescaled "true" self-avoiding walks to the Tóth-Werner "true" self-repelling motion

TL;DR

The paper proves that the rescaled TSAW on converges to the TSRM of Tóth and Werner, establishing a functional limit theorem via a joint generalized Ray-Knight theorem for the rescaled edge local times and their merge/absorption points. The core method inverts the GRKT to deduce convergence of the entire TSAW+local-time process, not merely finite-dimensional distributions, and relies on forward/backward Ray-Knight curves and Brownian-web-type structure to control coalescence events. This yields a robust framework for proving scaling limits of self-interacting walks by leveraging joint Ray-Knight representations and urn-process couplings. The results illuminate a general pathway for deriving process-level limits for self-interacting random walks whenever a GRKT is available, with potential applicability to generalized TSAW models and related edge-local-time processes.

Abstract

We prove that the rescaled ``true'' self-avoiding walk converges weakly as goes to infinity to the ``true'' self-repelling motion constructed by Tóth and Werner. The proof features a joint generalized Ray-Knight theorem for the rescaled local times processes and their merge and absorption points as the main tool for showing both the tightness and convergence of the finite dimensional distributions. Thus, our result can be seen as an example of establishing a functional limit theorem for a family of processes by inverting the joint generalized Ray-Knight theorem.

Paper Structure

This paper contains 17 sections, 25 theorems, 147 equations, 2 figures.

Table of Contents

  1. Introduction and the main result
  2. Preliminaries, the joint GRKT and its consequences
  3. Properties of TSRM and its local time
  4. Discrete Ray-Knight curves and the joint GRKT
  5. Consequences of the joint GRKT
  6. Proof of Theorem \ref{['thm:TSAW-FLT']}
  7. Proof of Theorem \ref{['thm:TSAW-FLL']}
  8. Joint generalized Ray-Knight Theorems (GRKTs)
  9. Tightness
  10. Associated urn processes
  11. Connection with directed edge local times
  12. Stationary distribution and convergence
  13. Relating the backward and the forward paths
  14. Gambler's ruin estimates
  15. Preliminaries
  16. ...and 2 more sections

Key Result

Theorem 1

Let $(\mathfrak{X}(\cdot),\mathfrak{H}(\cdot))$ be respectively the TSRM and its local time at the current location constructed in TW98. Define Then, as $n\to\infty$, the sequence of processes converges weakly in the standard Skorokhod topology on ${\mathcal{D}}^2_+$ to $(\mathfrak{X}(\cdot),\mathfrak{H}(\cdot))$.

Figures (2)

  • Figure 1: A reflected/absorbed two-sided Brownian motion started from $(x,h)$ with $x<0$ . It is reflected on the interval $[x,0]$ and absorbed on $\mathbb R\setminus[x,0]$. $T(x,h)$ is the area between the curve and the horizontal axis.
  • Figure 3: A visual depiction of the event $A^{(K),+}_{x,h,\delta,\varepsilon,n}$. The black curve in the middle depicts $\Lambda_{x,h}^n(\cdot)$. The short red curves depict pieces of the curves $\Lambda_{k\delta,\ell_k \sqrt{\delta}}^n(\cdot)$ and $\Lambda_{k\delta,(\ell_k+1)\sqrt{\delta}}^n(\cdot)$ which on the event $A^{(K),+}_{x,h,\delta,\varepsilon,n}$ do not meet.

Theorems & Definitions (53)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 4: Joint GRKT
  • Remark 5
  • Corollary 6
  • proof
  • Proposition 7
  • Theorem 8
  • Theorem 9: Joint weak limit at independent geometric stopping times
  • ...and 43 more