[Paper Review] Hypergeometric SLE with $κ=8$: Convergence of UST and LERW in Topological Rectangles

Yong Han, Mingchang Liu+1 more|arXiv (Cornell University)|Aug 2, 2020

Stochastic processes and statistical mechanics27 references4 citations

TL;DR

This paper establishes the convergence of the uniform spanning tree (UST) Peano curve and loop-erased random walk (LERW) in topological rectangles with alternating boundary conditions to hypergeometric SLE with $κ=8$, denoted $χ\text{SLE}_8$. It proves the continuity and reversibility of $χ\text{SLE}_8$, identifies its connection to standard $χ\text{SLE}_8$, and derives the limiting joint distribution of LERW endpoints via discrete holomorphic observables and tightness arguments.

ABSTRACT

We consider uniform spanning tree (UST) in topological rectangles with alternating boundary conditions. The Peano curves associated to the UST converge weakly to hypergeometric SLE$_8$, denoted by hSLE$_8$. From the convergence result, we obtain the continuity and reversibility of hSLE$_8$ as well as an interesting connection between SLE$_8$ and hSLE$_8$. The loop-erased random walk (LERW) branch in the UST converges weakly to SLE$_2(-1, -1; -1, -1)$. We also obtain the limiting joint distribution of the two end points of the LERW branch.

Motivation & Objective

• To establish the scaling limit of the UST Peano curve in topological rectangles with alternating boundary conditions.
• To prove the convergence of the UST Peano curve to $χ\text{SLE}_8$ and analyze its continuity and reversibility.
• To study the limiting behavior of the LERW branch in the UST and derive its joint distribution of endpoints.
• To extend the theory of hypergeometric SLE to the critical case $κ=8$, which was previously excluded due to technical challenges.
• To connect discrete holomorphic observables in lattice models to the continuous $χ\text{SLE}_8$ process.

Proposed method

• Use of discrete holomorphic observables to establish tightness of UST and LERW in quads.
• Application of conformal invariance and convergence of discrete harmonic functions to identify scaling limits.
• Construction of the $χ\text{SLE}_8$ process via a stochastic differential equation with drift involving hypergeometric functions.
• Proof of continuity and reversibility of $χ\text{SLE}_8$ by analyzing the SDE dynamics and boundary behavior.
• Derivation of the limiting joint distribution of LERW endpoints using convergence of random point pairs and martingale techniques.
• Comparison of $χ\text{SLE}_8$ with standard $χ\text{SLE}_8$ via parameter correspondence and SDE structure.

Experimental results

Research questions

• RQ1 Does the Peano curve of the uniform spanning tree in a topological rectangle with alternating boundary conditions converge weakly to a hypergeometric SLE process with $κ=8$?
• RQ2 Is the $χ\text{SLE}_8$ process continuous and reversible, and how does it differ from $χ\text{SLE}_\kappa$ for $κ>8$?
• RQ3 What is the limiting joint distribution of the two endpoints of the LERW branch in the UST on a lattice quad?
• RQ4 How does $χ\text{SLE}_8$ relate to standard $χ\text{SLE}_8$ in terms of driving function and SDE structure?
• RQ5 Can the convergence of UST and LERW be established using discrete holomorphic observables and tightness arguments in the quad setting?

Key findings

• The Peano curve of the UST in a topological rectangle with alternating boundary conditions converges weakly to $χ\text{SLE}_8$.
• The $χ\text{SLE}_8$ process is continuous and reversible, in contrast to $χ\text{SLE}_\kappa$ for $κ>8$, which is non-reversible.
• The LERW branch in the UST converges weakly to $χ\text{SLE}_2(-1,-1;-1,-1)$, a specific variant of hypergeometric SLE.
• The limiting joint distribution of the LERW endpoints is derived and shown to be non-uniform, reflecting the geometry of the quad.
• The $χ\text{SLE}_8$ process is connected to standard $χ\text{SLE}_8$ via a parameter correspondence in the SDE drift term.
• The convergence results are established via tightness of discrete processes and the use of discrete holomorphic observables to identify the scaling limit.