Universal Amplitude Ratios for Constrained Critical Systems

The critical properties of systems under constraint differ from their ideal counterparts through Fisher renormalization. The mathematical properties of Fisher renormalization applied to critical exponents are well known: the renormalized indices obey the same scaling relations as the ideal ones and the transformations are involutions in the sense that re-renormalizing the critical exponents of the constrained system delivers their original, ideal counterparts. Here we examine Fisher renormalization of critical amplitudes and show that, unlike for critical exponents, the associated transformations are not involutions. However, for ratios and combinations of amplitudes which are universal, Fisher renormalization is involutory.Comment: JSTAT published versio

Exact finite-size corrections for the spanning-tree model under different boundary conditions

We express the partition functions of the spanning tree on finite square lattices under five different sets of boundary conditions (free, cylindrical, toroidal, M\"obius strip, and Klein bottle) in terms of a principal partition function with twisted boundary conditions. Based on these expressions, we derive the exact asymptotic expansions of the logarithm of the partition function for each case. We have also established several groups of identities relating spanning-tree partition functions for the different boundary conditions. We also explain an apparent discrepancy between logarithmic correction terms in the free energy for a two dimensional spanning tree model with periodic and free boundary conditions and conformal field theory predictions. We have obtain corner free energy for the spanning tree under free boundary conditions in full agreement with conformal field theory predictions.Comment: 13 pages. Expanded text with additional result

Exact Solution of a Monomer-Dimer Problem: A Single Boundary Monomer on a Non-Bipartite Lattice

We solve the monomer-dimer problem on a non-bipartite lattice, the simple quartic lattice with cylindrical boundary conditions, with a single monomer residing on the boundary. Due to the non-bipartite nature of the lattice, the well-known method of a Temperley bijection of solving single-monomer problems cannot be used. In this paper we derive the solution by mapping the problem onto one on close-packed dimers on a related lattice. Finite-size analysis of the solution is carried out. We find from asymptotic expansions of the free energy that the central charge in the logarithmic conformal field theory assumes the value $c=-2$.Comment: 15 pages, 1 figure, submitted to Phy. Rev. E; v2: revised Acknowledgment

Exact phase diagrams for an Ising model on a two-layer Bethe lattice

Using an iteration technique, we obtain exact expressions for the free energy and the magnetization of an Ising model on a two - layer Bethe lattice with intralayer coupling constants J1 and J2 for the first and the second layer, respectively, and interlayer coupling constant J3 between the two layers; the Ising spins also couple with external magnetic fields, which are different in the two layers. We obtain exact phase diagrams for the system.Comment: 24 pages, 2 figures. To be published in Phys. Rev. E 59, Issue 6, 199