Scaling limit of the triangular kinetic prudent walk

TL;DR

This paper analyzes the scaling limit of the triangular kinetic prudent walk on the triangular lattice $\textbf{T}$, extending prior square-lattice results to a lattice with threefold symmetry. It develops a triangular Green's function $G_{\textbf{T}}(\lambda)$ via slab measures and a Radon–Nikodym framework, and proves that the scaled path converges to isotropic Brownian motion with covariance $\frac{1}{2}I$, under a normalization constant $\mathscr{L}$. Key innovations include a two- or three-degree-of-freedom decomposition, a Green's-function representation tied to slab measures, and careful comparison with previous BFV scaling results. The findings illuminate how lattice geometry shapes scaling constants and effective-limit descriptions, with potential implications for universality classes of prudent-like stochastic processes.

Abstract

Expressions for scaling limits of random walks, such as those obtained in several areas of the Probability theory literature, are of great significance in characterizing long term, stationary behavior of random processes. Presumably, in the limit of extremely long periods of time random walks and other stochastic processes including percolation are expected to fall into several \textit{universality classes}; such classes are of great significance in being able to disregard local, microscopic, dynamics, in favor of similar stationary dynamics in the bulk over long times. While several probabilistic models of interest are expected to behave similarly with respect to time, differences in fluctuations of limit shapes, and other stationary profiles, are expected to emerge. Given expressions obtained for the scaling limit of the kinetic prudent walk that have been obtained by Beffara, Friedli, and Velenik, in addition to the scaling limit for the kinetic uniform walk that have been obtained by Petrelis, Sun and Torri, we answer several questions pertaining to limit shapes, for the triangular prudent walk. While one would expect that the kinetic prudent walks, either over the square or triangular lattices, would have differing limit shapes, determining whether any differences between the scaling limit normalizations, and related quantities, emerge is of interest. In comparison to previously obtained scaling limits, we incorporate symmetries of the triangular lattice for computing the scale limit, beginning with a suitable basis.