A (2+1)-dimensional Gaussian field as fluctuations of quantum random walks on quantum groups

TL;DR

This paper constructs a (2+1)-dimensional Gaussian field describing fluctuations in quantum random walks on quantum groups, connecting it to Gaussian free fields and random matrix eigenvalues in specific sections.

Contribution

It introduces a novel (2+1)-dimensional Gaussian field linked to quantum random walks on quantum groups, extending previous models with a q-deformation.

Findings

Matches Gaussian free field on space-like paths

Corresponds to eigenvalues of random matrices

Describes asymptotic fluctuations in quantum walks

Abstract

This paper introduces a (2+1)-dimensional Gaussian field which has the Gaussian free field on the upper half-plane with zero boundary conditions as certain two-dimensional sections. Along these sections, called space-like paths, it matches the Gaussian field from eigenvalues of random matrices and from a growing random surface. However, along time-like paths the behavior is different. The Gaussian field arises as the asymptotic fluctuations in quantum random walks on quantum groups U_q(gl_n). This quantum random walk is a q-deformation of previously considered quantum random walks. When restricted to the space-like paths, the moments of the quantum random walk match the moments of the growing random surface.