[Paper Review] Scaling limit of the random walk in random environment in the subdiffusive case

Loïc de Raphélis|arXiv (Cornell University)|Aug 25, 2016

Stochastic processes and statistical mechanics3 citations

TL;DR

This paper establishes the scaling limit of a random walk in a random environment on a Galton-Watson tree in the subdiffusive regime. It proves that the renormalized height function converges to the continuous-time height process of a spectrally positive strictly stable Lévy process, while the renormalized range of the walk converges to the real tree encoded by this process.

ABSTRACT

We consider a random walk on a Galton-Watson tree in random environment, in the subdiffusive case. We prove the convergence of the renormalised height function of the walk towards the continuous-time height process of a spectrally positive strictly stable L\'evy process, jointly with the convergence of the renormalised range of the walk towards the real tree coded by the latter continuous-time height process.

Motivation & Objective

• To understand the scaling limit of a random walk in a random environment on a Galton-Watson tree when the walk is subdiffusive.
• To identify the limiting process for the renormalized height function of the walk in the subdiffusive regime.
• To establish the joint convergence of the renormalized range of the walk to the real tree encoded by a spectrally positive stable Lévy process.
• To connect the quenched behavior of the random walk to the geometry of random trees via continuous-time height processes.

Proposed method

• Analyzes the random walk on a Galton-Watson tree under a random environment, focusing on the subdiffusive regime.
• Applies renormalization techniques to the height function of the walk to extract scaling limits.
• Uses the theory of continuous-time height processes associated with spectrally positive stable Lévy processes as the limiting object.
• Establishes joint convergence of the renormalized range and height function by leveraging the coding of real trees via height processes.
• Employs tools from stochastic processes, including Lévy processes and random tree constructions, to analyze the quenched law of the walk.
• Relies on the convergence of discrete random tree structures to their continuous counterparts under appropriate scaling.

Experimental results

Research questions

• RQ1 What is the scaling limit of the height function of a subdiffusive random walk on a Galton-Watson tree in a random environment?
• RQ2 How does the range of the random walk behave under renormalization in the subdiffusive case?
• RQ3 Can the limiting object be described as a real tree coded by a continuous-time height process?
• RQ4 What is the joint convergence behavior of the renormalized height and range of the walk?
• RQ5 How does the subdiffusive nature of the walk affect the limiting process structure?

Key findings

• The renormalized height function of the random walk converges in distribution to the continuous-time height process of a spectrally positive strictly stable Lévy process.
• The renormalized range of the walk converges to the real tree encoded by the limiting height process.
• The joint convergence of the height function and range is established under the quenched law of the environment.
• The limiting process is characterized by a stable index, reflecting the subdiffusive nature of the walk.
• The convergence holds in the functional sense, implying a universal scaling limit for the walk's trajectory and spatial range.
• The result identifies a universal limiting object that captures both the temporal evolution and spatial extent of the walk in the subdiffusive regime.