[Paper Review] Scaling limit for a family of random paths with radial behavior

Cristian F. Coletti, León A. Valencia|arXiv (Cornell University)|Oct 25, 2013

Stochastic processes and statistical mechanics12 references3 citations

TL;DR

This paper introduces the Discrete Radial Poissonian Web (DRPW), a system of coalescing random paths with radial structure on nested circles in the plane. Under diffusive scaling, the DRPW converges in distribution to a restricted version of the Brownian Web—specifically, the T-Brownian Web—demonstrating weak convergence via a novel application of the FKG inequality to verify key convergence conditions.

ABSTRACT

We introduce a system of coalescing random paths with radialbehavior in a subsetof the plane. We call it theDiscrete Radial Poissonian Web. We show that underdiffusive scaling this family converges in distribution toa mapping of a restrictionof the Brownian Web.

Motivation & Objective

• To define and formalize a new stochastic process—Discrete Radial Poissonian Web (DRPW)—with radial coalescing paths on concentric circles.
• To establish the weak convergence of the DRPW to a restricted form of the Brownian Web (the T-Brownian Web) under diffusive scaling.
• To verify the convergence criteria (I, B₁, B₂) for the Brownian Web using a novel application of the FKG inequality to handle path coalescence in radial geometry.
• To demonstrate that a simpler radial model, despite lacking full radial spanning tree structure, still converges to the Brownian Bridge Web, providing a new construction for this limiting object.

Proposed method

• Define a sequence of nested circles of radius $ n, n-1, ..., 1 $, each equipped with an independent Poisson point process of rate 1 on the circumference.
• Construct paths from each point on a circle to the nearest point on the next inner circle, with fallback rules if no point exists within a specified angular window.
• Introduce spatial restrictions $ ilde{ heta}_n $ and $ ilde{ ho}_n $ to control angular spread and ensure path coalescence behavior.
• Apply a diffusive scaling limit by rescaling space and time, mapping the discrete paths to continuous trajectories.
• Verify convergence using a modified version of the convergence criteria from [9], focusing on conditions I (tightness), B₁ (path existence), and B₂ (coalescence behavior), with FKG inequality used to control path interactions.
• Prove asymptotic tail behavior of coalescence times and use Lindeberg’s condition to establish Gaussian limits for increments, supporting weak convergence.

Experimental results

Research questions

• RQ1 Does a system of coalescing random paths with radial structure converge to a known universal limit in the diffusive scaling limit?
• RQ2 Can the FKG inequality be effectively applied to verify the B₂ condition in a radial, non-uniform spatial setting?
• RQ3 How does the radial geometry affect the coalescence time distribution and the limiting behavior of the path system?
• RQ4 Is the DRPW a valid construction that converges to the Brownian Bridge Web, despite being simpler than prior models?
• RQ5 What is the role of angular and radial restrictions in ensuring convergence to a well-defined limiting object?

Key findings

• The DRPW converges in distribution to the T-Brownian Web under diffusive scaling, establishing a new construction for the Brownian Bridge Web.
• The FKG inequality is successfully applied to verify condition B₂, which controls the coalescence behavior of paths in the radial framework.
• The coalescence time between two typical paths in the DRPW has a tail that decays faster than any polynomial, supporting tightness and convergence.
• The limiting process exhibits Gaussian increment behavior, with variance proportional to $ g(t) = rac{1}{1-t} $, confirming the scaling limit's consistency with the Brownian Web framework.
• The convergence is established via a full verification of the convergence criteria I, B₁, and B₂, with B₁ and I verified through path existence and moment conditions.
• The result holds even without full radial spanning tree structure, showing that radial coalescence with local nearest-neighbor rules suffices for convergence to the Brownian Bridge Web.