A model of loops in 2-D

TL;DR

This paper investigates the dynamical properties of a 2D loop model derived from the gonihedric spin model, combining Monte Carlo simulations with analytical solutions of matrix models to understand its behavior.

Contribution

It provides the first numerical study of the 2D loop model's dynamics and offers an analytical approach by reducing a two-matrix model to a solvable one-matrix model.

Findings

Monte Carlo simulations reveal trivial thermodynamical properties in 2D.

Analytical reduction simplifies the two-matrix model to a one-matrix model.

Partial solutions to the coupled gravity model are obtained.

Abstract

The gonihedric spin model was first introduced as the action for a discretized tensionless string in a discretized embeding space. Afterwards was found that there are interesting features on the dynamical behavior of this model in 3 dimensions (as it was first formulated) that make us think on glassy spin model without inherent disorder. Extensive simulations have been carried out in the 3 dimensional model. In the following I will report on a work composed of two different but related parts. The first part is a numerical study through Monte Carlo simulations of the dynamical properties of the 2 dimensional version of the model (i.e. the loop model), which is much simpler due to the fact that it has trivial thermodynamical properties. The second part consists on an analytical approach of this 2 dimensional loop model coupled to gravity. We solve partially the associated two-matrix model via a reduction to an equivalent one matrix model and saddle point methods with the last one-matrix model.