[Paper Review] Topology of the support of the two-dimensional random walk

Frédéric van Wijland, S. Caser+1 more|arXiv (Cornell University)|Sep 2, 1996

Stochastic processes and statistical mechanics1 references3 citations

TL;DR

This paper investigates the topological properties of the support (visited sites) of a two-dimensional simple symmetric random walk on a square lattice in the long-time limit. It shows that all global observables of the support—such as the number of visited sites, boundary length, and number of enclosed unvisited islands—scale as t/log^k t, and their fluctuations are universally proportional to a single random process η(t), which converges to Varadhan’s renormalized local time of self-intersections, implying all fluctuations scale with the area fluctuations ΔS(t).

ABSTRACT

We study the support (i.e. the set of visited sites) of a t step random walk on a two-dimensional square lattice in the large t limit. A broad class of global properties M(t) of the support is considered, including, e.g., the number S(t) of its sites; the length of its boundary; the number of islands of unvisited sites that it encloses; the number of such islands of given shape, size, and orientation; and the number of occurrences in space of specific local patterns of visited and unvisited sites. On a finite lattice we determine the scaling functions that describe the averages on appropriate lattice size dependent time scales. On an infinite lattice we first observe that the all increase with t as t/\log^k t, where k is an M dependent positive integer. We then consider the class of random processes constituted by the fluctuations around average Delta M(t). We show that to leading order as t gets large these fluctuations are all proportional to a single universal random process eta(t), normalized to =1$. For t--> infinity the probability law of eta(t) tends to that of Varadhan's renormalized local time of self-intersections. An implication is that in the long time limit all Delta M(t) are proportional to Delta S(t).

Motivation & Objective

• To characterize the large-time behavior of topological properties of the support of a 2D simple symmetric random walk on a square lattice.
• To determine the scaling behavior of global observables M(t), including site count, boundary length, and number of enclosed unvisited islands.
• To analyze the statistical fluctuations ΔM(t) around the average <M(t)> and identify universal scaling features.
• To establish a universal connection between the fluctuations of different support observables via a single random process η(t).

Proposed method

• Analyzes the support of a t-step simple symmetric random walk on a 2D square lattice in the limit t → ∞.
• Considers a broad class of global observables M(t), such as S(t) (number of visited sites), boundary length, number of enclosed unvisited islands, and local site patterns.
• On a finite lattice, determines scaling functions for <M(t)> on lattice-size-dependent time scales.
• On an infinite lattice, derives the leading-order asymptotic scaling <M(t)> ∼ t / log^k t, where k depends on the observable M.
• Analyzes fluctuations ΔM(t) = M(t) − <M(t)>, showing they are all proportional to a single universal random process η(t) with ⟨η²(t)⟩ = 1.
• Demonstrates that as t → ∞, the probability law of η(t) converges to that of Varadhan’s renormalized local time of self-intersections of the walk.

Experimental results

Research questions

• RQ1 How do global topological properties of the support of a 2D random walk scale with time t in the large t limit?
• RQ2 What is the nature of the fluctuations ΔM(t) around the average values of these support observables?
• RQ3 Is there a universal random process that governs the fluctuations of all such observables?
• RQ4 How is the universal fluctuation process related to known quantities in self-intersection theory, such as Varadhan’s renormalized local time?
• RQ5 To what extent are the fluctuations of different support observables, such as the number of visited sites and the number of enclosed islands, correlated?

Key findings

• All global observables M(t) of the support scale asymptotically as t / log^k t, where k is a positive integer depending on the observable.
• The fluctuations ΔM(t) of all such observables are universally proportional to a single random process η(t), normalized to have unit variance.
• As t → ∞, the probability distribution of η(t) converges to that of Varadhan’s renormalized local time of self-intersections of the random walk.
• This convergence implies that all ΔM(t) are asymptotically proportional to ΔS(t), the fluctuation in the number of visited sites.
• The universal scaling of fluctuations establishes a deep connection between the topology of the support and the self-intersection statistics of the walk.
• The results hold both on finite lattices (with appropriate time scaling) and on the infinite lattice in the long-time limit.