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An exact universal amplitude ratio for percolation

Katherine Seaton

Published Jan 1, 2001• 2 citations
OpenAlex

Abstract

The universal amplitude ratio $\\tilde{R}_{\\xi}$ for percolation in two dimensions is determined exactly using results for the dilute A model in regime 1, by way of a relationship with the q-state Potts model for q<4. The numerical result obtained by Delfino and Cardy by quantum field theory techniques agrees well with the algebraic expression determined here.

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The universal amplitude ratio tildeRxi\\tilde{R}_{\\xi}tildeRxi​ for percolation in two dimensions is determined exactly using results for the dilute A model in regime 1, by way of a relationship with the q-state Potts model for q<4. The numerical result obtained by Delfino and Cardy by quantum field theory techniques agrees well with the algebraic expression determined here.
Universal amplitude ratios in the two-dimensional qqq-state Potts model and percolation from quantum field theory
We consider the scaling limit of the two-dimensional qqq-state Potts model for qleq4q\\leq 4qleq4. We use the exact scattering theory proposed by Chim and Zamolodchikov to determine the one and two-kink form factors of the energy, order and disorder operators in the model. Correlation functions and universal combinations of critical amplitudes are then computed within the two-kink approximation in the form factor approach. Very good agreement is found whenever comparison with exact results is possible. We finally consider the limit qto1q\\to 1qto1 which is related to the isotropic percolation problem. Although this case presents a serious technical difficulty, we predict a value close to 74 for the ratio of the mean cluster size amplitudes above and below the percolation threshold. Previous estimates for this quantity range from 14 to 220.
The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.