Exact Spectral Dimension of the Random Surface

TL;DR

This paper introduces a new analytical method to compute the spectral dimension of a random surface, demonstrating it equals two by linking random walks to the q-state Potts model and using matrix model techniques.

Contribution

The paper presents a novel analytical approach connecting random walk spectral dimension to the q-state Potts model on random surfaces, providing a precise calculation.

Findings

Spectral dimension of the random surface is exactly two.

The method uses the equivalence of random walk and Potts model with non-zero magnetic field.

Matrix model techniques are employed to compute the critical exponent.

Abstract

We propose a new method of the analytical computation of the spectral dimension which is based on the equivalence of the random walk and the q-state Potts model with non-zero magnetic field in the limit $q\to 0$. Calculating the critical exponent of the magnetization of this model on the dynamically triangulated random surface by means of a matrix model technique we obtain that the spectral dimension of this surface is equal to two.