Percolation on Strings and the Cover-up of the c=1 Disaster

TL;DR

This paper investigates percolation on string theory worldsheets across different central charges, revealing how finite-size effects and log corrections influence numerical results and potentially obscure the true continuum behavior for higher c values.

Contribution

It provides a detailed analysis of percolation on string worldsheets, especially at c=1, and discusses how finite-size effects impact the detection of critical phenomena at higher c.

Findings

Agreement of simulation exponents with theoretical values for c<1

Log corrections reconcile numerical results with KPZ predictions at c=1

Finite-size effects likely obscure continuum behavior for c>1

Abstract

We study percolation on the worldsheets of string theory for $c=0,1/2,1$ and $2$. For $c<1$ we find that critical exponents measured from simulations agree quite well with the theoretical values. For $c=1$ we show how log corrections determined from the exact solution reconcile numerical results with the KPZ predictions. We extend this analysis to the large $c$ regime and estimate how finite-size effects will effectively raise the ground state energy, masking the presence of the tachyon for moderate values of $c > 1$. It thus appears likely that simulations for $c=2,3 \ldots$ on numerically accessible lattices will fail to even capture the qualitative behavior of the continuum limit.