The scaling limit of fair Peano paths

TL;DR

This paper investigates the scaling limits of fair Peano paths derived from random spanning trees on planar grids, revealing a deterministic limit contrasting with the known stochastic limit for uniform trees.

Contribution

It demonstrates that fair Peano paths have a deterministic scaling limit, extending understanding of path behavior in random spanning trees beyond uniform cases.

Findings

Fair Peano paths have a deterministic scaling limit.

Contrasts with the stochastic SLE_8 limit for uniform trees.

Provides insights into the effect of fairness constraints on path scaling.

Abstract

We study random Peano paths on planar square grids that arise from fair random spanning trees. These are trees that are sampled in such a way as to have the same (if possible) edge probabilities. In particular, we are interested in identifying the scaling limit as the mesh-size of the grid tends to zero. It is known \cite{lawler-schramm-werner2002} that if the trees are sampled uniformly, then the scaling limit exists and equals ${\rm SLE}_8$. We show that if we simply follow the same steps as in \cite{lawler-schramm-werner2002}, then fair Peano paths have a deterministic scaling limit.