Critical Percolation in Finite Geometries

The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent.Comment: 10 page

Critical behaviour in parabolic geometries

We study two-dimensional systems with boundary curves described by power laws. Using conformal mappings we obtain the correlations at the bulk critical point. Three different classes of behaviour are found and explained by scaling arguments which also apply to higher dimensions. For an Ising system of parabolic shape the behaviour of the order at the tip is also found.Comment: Old paper, for archiving. 6 pages, 1 figure, epsf, IOP macr

Fermionic field theory for directed percolation in (1+1) dimensions

We formulate directed percolation in (1+1) dimensions in the language of a reaction-diffusion process with exclusion taking place in one space dimension. We map the master equation that describes the dynamics of the system onto a quantum spin chain problem. From there we build an interacting fermionic field theory of a new type. We study the resulting theory using renormalization group techniques. This yields numerical estimates for the critical exponents and provides a new alternative analytic systematic procedure to study low-dimensional directed percolation.Comment: 20 pages, 2 figure

The Number of Incipient Spanning Clusters in Two-Dimensional Percolation

Using methods of conformal field theory, we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k, in the limit when k is large.Comment: 9 pages, LaTeX, 1 eps figure. Additional references and comparison with existing numerical results include

Critical phenomena and quantum phase transition in long range Heisenberg antiferromagnetic chains

Antiferromagnetic Hamiltonians with short-range, non-frustrating interactions are well-known to exhibit long range magnetic order in dimensions, $d\geq 2$ but exhibit only quasi long range order, with power law decay of correlations, in d=1 (for half-integer spin). On the other hand, non-frustrating long range interactions can induce long range order in d=1. We study Hamiltonians in which the long range interactions have an adjustable amplitude lambda, as well as an adjustable power-law $1/|x|^\alpha$, using a combination of quantum Monte Carlo and analytic methods: spin-wave, large-N non-linear sigma model, and renormalization group methods. We map out the phase diagram in the lambda-alpha plane and study the nature of the critical line separating the phases with long range and quasi long range order. We find that this corresponds to a novel line of critical points with continuously varying critical exponents and a dynamical exponent, z<1.Comment: 27 pages, 12 figures. RG flow added. Final version to appear in JSTA

Long-range epidemic spreading with immunization

We study the phase transition between survival and extinction in an epidemic process with long-range interactions and immunization. This model can be viewed as the well-known general epidemic process (GEP) in which nearest-neighbor interactions are replaced by Levy flights over distances r which are distributed as P(r) ~ r^(-d-sigma). By extensive numerical simulations we confirm previous field-theoretical results obtained by Janssen et al. [Eur. Phys. J. B7, 137 (1999)].Comment: LaTeX, 14 pages, 4 eps figure

Strong Conformal Dynamics at the LHC and on the Lattice

Conformal technicolor is a paradigm for new physics at LHC that may solve the problems of strong electroweak symmetry breaking for quark masses and precision electroweak data. We give explicit examples of conformal technicolor theories based on a QCD-like sector. We suggest a practical method to test the conformal dynamics of these theories on the lattice.Comment: v2: Generalized discussion of lattice measurement of hadron masses, references added, minor clarifications v3: references added, minor change

Exact Partition Function and Boundary State of 2-D Massive Ising Field Theory with Boundary Magnetic Field

We compute the exact partition function, the universal ground state degeneracy and boundary state of the 2-D Ising model with boundary magnetic field at off-critical temperatures. The model has a domain that exhibits states localized near the boundaries. We study this domain of boundary bound state and derive exact expressions for the ``$g$ function'' and boundary state for all temperatures and boundary magnetic fields. In the massless limit we recover the boundary renormalization group flow between the conformally invariant free and fixed boundary conditions.Comment: plain latex, 17 pages plus 11 figures in 3 .ps files, uuencoded in isfig.u

A crossing probability for critical percolation in two dimensions

Langlands et al. considered two crossing probabilities, pi_h and pi_{hv}, in their extensive numerical investigations of critical percolation in two dimensions. Cardy was able to find the exact form of pi_h by treating it as a correlation function of boundary operators in the Q goes to 1 limit of the Q state Potts model. We extend his results to find an analogous formula for pi_{hv} which compares very well with the numerical results.Comment: 8 pages, Latex2e, 1 figure, uuencoded compressed tar file, (1 typo changed

Discrete Holomorphicity at Two-Dimensional Critical Points

After a brief review of the historical role of analyticity in the study of critical phenomena, an account is given of recent discoveries of discretely holomorphic observables in critical two-dimensional lattice models. These are objects whose correlation functions satisfy a discrete version of the Cauchy-Riemann relations. Their existence appears to have a deep relation with the integrability of the model, and they are presumably the lattice versions of the truly holomorphic observables appearing in the conformal field theory (CFT) describing the continuum limit. This hypothesis sheds light on the connection between CFT and integrability, and, if verified, can also be used to prove that the scaling limit of certain discrete curves in these models is described by Schramm-Loewner evolution (SLE).Comment: Invited talk at the 100th Statistical Mechanics Meeting, Rutgers, December 200