Critical Exponents from General Distributions of Zeroes

All of the thermodynamic information on a statistical mechanical system is encoded in the locus and density of its partition function zeroes. Recently, a new technique was developed which enables the extraction of the latter using finite-size data of the type typically garnered from a computational approach. Here that method is extended to deal with more general cases. Other critical points of a type which appear in many models are also studied.Comment: 4 pages, 3 figure

Anomalous scaling and Lee-Yang zeroes in Self-Organized Criticality

We show that the generating functions of avalanche observables in SOC models exhibits a Lee-Yang phenomenon. This establishes a new link between the classical theory of critical phenomena and SOC. A scaling theory of the Lee-Yang zeroes is proposed including finite sampling effects.Comment: 33 pages, 19 figures, submitte

The Strength of First and Second Order Phase Transitions from Partition Function Zeroes

We present a numerical technique employing the density of partition function zeroes (i) to distinguish between phase transitions of first and higher order, (ii) to examine the crossover between such phase transitions and (iii) to measure the strength of first and second order phase transitions in the form of latent heat and critical exponents. These techniques are demonstrated in applications to a number of models for which zeroes are available.Comment: 18 pages, LaTeX, 6 postscript figures, accepted for publication in J. Stat. Phy

A study of the phase transition in 4D pure compact U(1) LGT on toroidal and spherical lattices

We have performed a systematic study of the phase transition in the pure compact U(1) lattice gauge theory in the extended coupling parameter space (\beta, \gamma) on toroidal and spherical lattices. The observation of a non-zero latent heat in both topologies for all investigated \gamma in the interval [+0.2,-0.4], together with an effective exponent \nu = 1/d when large enough lattices are considered, lead us to conclude that the phase transition is first order. For negative \gamma, our results point to an increasingly weak first order transition as \gamma is made more negative.Comment: References added; To appear in Nuc. Phys.

Universal amplitude ratios and Coxeter geometry in the dilute A model

The leading excitations of the dilute $A_L$ model in regime 2 are considered using analytic arguments. The model can be identified with the integrable $\phi_{1,2}$ perturbation of the unitary minimal series $M_{L,L+1}$. It is demonstrated that the excitation spectrum of the transfer matrix satisfies the same functional equations in terms of elliptic functions as the exact S-matrices of the $\phi_{1,2}$ perturbation do in terms of trigonometric functions. In particular, the bootstrap equation corresponding to a self-fusing process is recovered. For the special cases $L=3,4,6$ corresponding to the Ising model in a magnetic field, and the leading thermal perturbations of the tricritical Ising and three-state Potts model, as well as for the unrestricted model, $L=\infty$, we relate the structure of the Bethe roots to the Lie algebras $E_{8,7,6}$ and $D_4$ using Coxeter geometry. In these cases Coxeter geometry also allows for a single formula in generic Lie algebraic terms describing all four cases. For general $L$ we calculate the spectral gaps associated with the leading excitation which allows us to compute universal amplitude ratios characteristic of the universality class. The ratios are of field theoretic importance as they enter the bulk vacuum expectation value of the energy momentum tensor associated with the corresponding integrable quantum field theories.Comment: 32 pages (tcilatex

Properties of phase transitions of higher order

There is only limited experimental evidence for the existence in nature of phase transitions of Ehrenfest order greater than two. However, there is no physical reason for their non-existence, and such transitions certainly exist in a number of theoretical models in statistical physics and lattice field theory. Here, higher-order transitions are analysed through the medium of partition function zeros. Results concerning the distributions of zeros are derived as are scaling relations between some of the critical exponents.Comment: 6 pages, poster presented at Lattice 2005 (Spin and Higgs), Trinity College Dubli

New Multicritical Random Matrix Ensembles

In this paper we construct a class of random matrix ensembles labelled by a real parameter $\alpha \in (0,1)$, whose eigenvalue density near zero behaves like $|x|^\alpha$. The eigenvalue spacing near zero scales like $1/N^{1/(1+\alpha)}$ and thus these ensembles are representatives of a {\em continous} series of new universality classes. We study these ensembles both in the bulk and on the scale of eigenvalue spacing. In the former case we obtain formulas for the eigenvalue density, while in the latter case we obtain approximate expressions for the scaling functions in the microscopic limit using a very simple approximate method based on the location of zeroes of orthogonal polynomials.Comment: 15 pages, 3 figures; v2: version to appear in Nucl. Phys.

Level-spacing distribution of a fractal matrix

We diagonalize numerically a Fibonacci matrix with fractal Hilbert space structure of dimension $d_{f}=1.8316...$ We show that the density of states is logarithmically normal while the corresponding level-statistics can be described as critical since the nearest-neighbor distribution function approaches the intermediate semi-Poisson curve. We find that the eigenvector amplitudes of this matrix are also critical lying between extended and localized.Comment: 6 pages, Latex file, 4 postscript files, published in Phys. Lett. A289 pp 183-7 (2001

Self-Organized Criticality and Thermodynamic formalism

We introduce a dissipative version of the Zhang's model of Self-Organized Criticality, where a parameter allows to tune the local energy dissipation. We analyze the main dynamical features of the model and relate in particular the Lyapunov spectrum with the transport properties in the stationary regime. We develop a thermodynamic formalism where we define formal Gibbs measure, partition function and pressure characterizing the avalanche distributions. We discuss the infinite size limit in this setting. We show in particular that a Lee-Yang phenomenon occurs in this model, for the only conservative case. This suggests new connexions to classical critical phenomena.Comment: 35 pages, 15 Figures, submitte