Finite Size Effects for the Ising Model Coupled to 2-D Random Surfaces

TL;DR

This paper investigates finite size effects in the Ising model coupled to 2D random surfaces using exact 2-matrix model results, developing new algorithms and analytic methods to analyze corrections and susceptibilities.

Contribution

It introduces an efficient recursive algorithm for solving quintic equations and develops an analytic approach based on singularity behavior to study finite size effects.

Findings

Finite size corrections differ from traditional phenomenological models.

The developed methods accurately analyze free energy near singularities.

Results have implications for numerical estimation of string susceptibility.

Abstract

Finite size effects for the Ising Model coupled to two dimensional random surfaces are studied by exploiting the exact results from the 2-matrix models. The fixed area partition function is numerically calculated with arbitrary precision by developing an efficient algorithm for recursively solving the quintic equations so encountered. An analytic method for studying finite size effects is developed based on the behaviour of the free energy near its singular points. The generic form of finite size corrections so obtained are seen to be quite different from the phenomenological parameterisations used in the literature. The method of singularities is also applied to study the magnetic susceptibility. A brief discussion is presented on the implications of these results to the problem of a reliable determination of string susceptibility from numerical simulations.