Anomalous 1D fluctuations of a simple 2D random walk in a large deviation regime

TL;DR

This paper investigates how two-dimensional random walks under large deviation constraints exhibit one-dimensional KPZ fluctuations, with the fluctuation behavior depending on the geometry of the imposed constraints.

Contribution

It demonstrates that 2D random walks in large deviation regimes can show KPZ-type fluctuations with geometry-dependent exponents, supported by heuristic, analytic, and numerical analysis.

Findings

KPZ fluctuations occur in 2D random walks constrained by geometry

The span scales as t^{1/3} for semicircular constraints

The span remains bounded (scales as t^0) for triangular constraints

Abstract

The following question is the subject of our work: could a two-dimensional random path pushed by some constraints to an improbable "large deviation regime", possess extreme statistics with one-dimensional Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though non-universal, since the fluctuations depend on the underlying geometry. We consider in details two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span $\left< d(t) \right> \sim t^{\gamma}$ of the trajectory of $t$ steps above the top of the semicircle or the triangle. We show that $\gamma = \frac{1}{3}$, i.e. $\left< d(t) \right>$ shares the KPZ statistics for the semicircle, while $\gamma=0$ for the triangle. We propose heuristic derivations of scaling exponents $\gamma$ for different geometries, justify them by explicit analytic computations and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.