Universality of the Brownian net

TL;DR

The Brownian net is shown to be the universal scaling limit for one-dimensional branching-coalescing random walks with weak binary branching under the diffusive scaling map $S_{\varepsilon,\sigma}$. The approach combines a lower bound via convergence to a pair of sticky Brownian webs and a hopping construction with two coupled webs, with an upper bound obtained by domination by a Bernoulli net and a multiscale tightness argument. Under finite $(3+\eta)$-th moment on the jump distribution, the rescaled discrete net $S_{\varepsilon,\sigma}N_\varepsilon$ converges to the standard Brownian net ${\cal N}$, matching the image of ${\cal N}$ under scaling and establishing universality in the crossing regime. The results provide a robust framework for analyzing crossing branching-coalescing systems and extend classical web convergence to the crossing setting, enabling applications to a broad class of interacting path systems.

Abstract

The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from every point in space and time, while the Brownian net is an extension that also allows branching. We show here that the Brownian net is the universal scaling limit of one-dimensional branching-coalescing random walks with weak binary branching and arbitrary increment distributions that have finite $(3+\varepsilon)$-th moment. This gives the first example in the domain of attraction of the Brownian net where paths can cross without coalescing.

Figures (5)

• Figure 1: A path $\pi$ causing the unlikely event $\{{\cal X}_n\in A_{M,\delta}(x,t)\}$ to occur.
• Figure 2: Two cases of $\varpi$ branching off from $\pi$ at $z_0$, but does not pass through the RBP $z_2$.
• Figure 3: The image on the left illustrates the exploration procedure for $\{{\cal R}_T\geq2\}$. The image on the right illustrates the branching-coalescing point set, with $\bullet$ identifying the relevant branching points (RBP's).
• Figure 4: Illustration of the $j$-th iteration of the multiscale argument.
• Figure 5: For random walks starting from $\xi^{(j-1)}_{2i-1}$ at time $T^{(j-1)}_{2i-1}$, Lemma \ref{['L:denbc']} is applied three times with $p=1/\sqrt{2}$ and successive time durations $T=2^{j-1}R_0$, $2T$, and $4T$.

Theorems & Definitions (45)

• Theorem 1.1: Convergence to the Brownian net
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 2.1
• Theorem 2.2: Characterization of the Brownian web
• Theorem 2.3: Convergence to the Brownian web
• Proposition 2.4: Characterization of sticky Brownian motions
• Theorem 2.5: Martingale characterization of sticky Brownian webs
• Proposition 2.6: Equivalence