I knew I should have taken that left turn at Albuquerque

TL;DR

This paper investigates the Laplacian-infinity path, revealing that unlike finite alpha cases, its scaling limit depends on lattice structure and lacks conformal invariance, challenging previous assumptions about convergence to SLE.

Contribution

It demonstrates that the Laplacian-infinity path does not converge to SLE and its limit is lattice-dependent, contrasting finite alpha cases.

Findings

Scaling limit depends on lattice structure

No conformal invariance in the limit

Contradicts expectations of convergence to SLE

Abstract

We study the Laplacian-infinity path as an extreme case of the Laplacian-alpha random walk. Although, in the finite alpha case, there is reason to believe that the process converges to SLE, we show that this is not the case when alpha is infinite. In fact, the scaling limit depends heavily on the lattice structure, and is not conformal (or even rotational) invariant.