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

2025/02/16 by Elena Kosygina, Kosygina, Elena, Jonathon Peterson +1 · 3 citations
Engineering · Physics and Astronomy · #Sports Dynamics and Biomechanics #Experimental and Theoretical Physics Studies #Micro and Nano Robotics

paper · pdf · doi:10.48550/arxiv.2502.10960

Abstract

We prove that the rescaled ``true'' self-avoiding walk (n<sup>-2/3</sup>X<sub>\lfloor nt \rfloor</sub>)<sub>t∈ℝ+</sub> 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.

Cited by

• Convergence rates of self-repellent random walks, their local time and Event Chain Monte Carlo
• Construction of the 1d Self-repelling Brownian Polymer
• "True" self-avoiding walks on general trees

Related

• Convergence and non-convergence of scaled self-interacting random walks to Brownian motion perturbed at extrema
• Large deviations and path properties of the true self-repelling motion
• Long-range self-avoiding walk converges to alpha-stable processes
• Diffusive limit for the myopic (or "true") self-avoiding random walk in three and more dimension
• Central Limit Theorem for the self-repelling random walk with directed edges