Integrable 2D Lorentzian Gravity and Random Walks

TL;DR

This paper introduces integrable models of 2D Lorentzian gravity with higher curvature, establishing a link to directed random walks and explaining their fractal properties and universality class.

Contribution

It presents a family of exactly solvable discrete models of 2D Lorentzian gravity with higher curvature and explores their relation to random walks and universality classes.

Findings

Models possess mutually commuting transfer matrices.

Spectral parameter interpolates between flat and curved space-times.

Lorentzian triangulations have fractal dimension 2 and belong to the pure Lorentzian gravity universality class.

Abstract

We introduce and solve a family of discrete models of 2D Lorentzian gravity with higher curvature, which possess mutually commuting transfer matrices, and whose spectral parameter interpolates between flat and curved space-times. We further establish a one-to-one correspondence between Lorentzian triangulations and directed Random Walks. This gives a simple explanation why the Lorentzian triangulations have fractal dimension 2 and why the curvature model lies in the universality class of pure Lorentzian gravity. We also study integrable generalizations of the curvature model with arbitrary polygonal tiles. All of them are found to lie in the same universality class.