Expansion for the critical point of site percolation: the first three terms

We expand the critical point for site percolation on the d-dimensional hypercubic lattice in terms of inverse powers of 2d, and we obtain the first three terms rigorously. This is achieved using the lace expansion.

An exact mapping between loop-erased random walks and an interacting field theory with two fermions and one boson

We give a simplified proof for the equivalence of loop-erased random walks to a lattice model containing two complex fermions, and one complex boson. This equivalence works on an arbitrary directed graph. Specifying to the dd-dimensional hypercubic lattice, at large scales this theory reduces to a scalar \phi^4ϕ4-type theory with two complex fermions, and one complex boson. While the path integral for the fermions is the Berezin integral, for the bosonic field we can either use a complex field \phi(x)\in \mathbb Cϕ(x)∈ℂ (standard formulation) or a nilpotent one satisfying \phi(x)^2 =0ϕ(x)2=0. We discuss basic properties of the latter formulation, which has distinct advantages in the lattice model.

Critical Site Percolation in High Dimension

We use the lace expansion to prove an infra-red bound for site percolation on the hypercubic lattice in high dimension. This implies the triangle condition and allows us to derive several critical exponents that characterize mean-field behavior in high dimensions.

Modeling of quasicrystal lattices with a 4-fold symmetry axis

The construction method of a quasilattice with a four-fold rotational symmetry axis is proposed. The described method is based on the recurrent generation of the initial group of lattice points, which are a set of vertices of a square. The aperiodic crystal reciprocal lattice modeling algorithm is analyzed. Used modeling technique is compared with conventional projection approach. The orthogonal basis of a four-dimensional hypercubic lattice is proposed. This lattice produces two-dimensional quasicrystal with a four-fold symmetry axis after it projection on a flat surface. It is shown that the indexation of diffraction pattern of similar quasiperiodic structures can be carry out using 3 integer indexes, which is analogous to the indexing system proposed by Cahn for application to icosahedral quasicrystals.

N = 2* Yang-Mills on the Lattice

The N = 2* Yang-Mills theory in four dimensions is a non-conformal theory that appears as a mass deformation of maximally supersymmetric N = 4 Yang-Mills theory. This theory also takes part in the AdS/CFT correspondence and its gravity dual is type IIB supergravity on the Pilch-Warner background. The finite temperature properties of this theory have been studied recently in the literature. It has been argued that at large N and strong coupling this theory exhibits no thermal phase transition at any nonzero temperature. The low temperature N = 2* plasma can be compared to the QCD plasma. We provide a lattice construction of N = 2* Yang-Mills on a hypercubic lattice starting from the N = 4 gauge theory. The lattice construction is local, gauge-invariant, free from fermion doubling problem and preserves a part of the supersymmetry. This nonperturbative formulation of the theory can be used to provide a highly nontrivial check of the AdS/CFT correspondence in a non-conformal theory.

A consistent measure for lattice Yang–Mills

The construction of a consistent measure for Yang–Mills is a precondition for an accurate formulation of nonperturbative approaches to QCD, both analytical and numerical. Using projective limits as subsets of Cartesian products of homomorphisms from a lattice to the structure group, a consistent interaction measure and an infinite-dimensional calculus have been constructed for a theory of non-Abelian generalized connections on a hypercubic lattice. Here, after reviewing and clarifying past work, new results are obtained for the mass gap when the structure group is compact.

Center vortices, area law and the catenary solution

We present meson–meson (Wilson loop) correlators in Z(2) center vortex models for the infrared sector of Yang–Mills theory, i.e. a hypercubic lattice model of random vortex surfaces and a continuous (2 + 1)-dimensional model of random vortex lines. In particular, we calculate quadratic and circular Wilson loop correlators in the two models, respectively, and observe that their expectation values follow the area law and show string breaking behavior. Further, we calculate the catenary solution for the two cases and try to find indications for minimal surface behavior or string surface tension leading to string constriction.