At infinite N, continuum Euclidean SU N gauge theory defined on a symmetrical four torus has a rich phase structure with phases where the finite volume system behaves as if it had infinite extent in some or all of the directions. In addition, fermions are automatically quenched, so planar QCD should be cheaper to solve numerically that full QCD. Large N is a relatively unexplored and worthwhile...

In this paper we study connectivity and hamiltonicity properties of the topological grid graphs, which are a natural type of planar graphs associated with finite subgraphs of the usual square lattice graph of the plane. The main results are as follows. The shortness coefficient of the family of all topological grid graphs is at most 16/17. Every 3-connected topological grid graph is hamiltonian.

Every convex set in the plane gives rise to geometric functionals such as the area, perimeter, diameter, width, inradius and circumradius. In this paper, we prove new inequalities involving these geometric functionals for planar convex sets containing zero or one interior lattice point. We also conjecture two results concerning sets containing one interior lattice point. Finally, we summarize k...

Several instances of distributive lattices on graph structures are known. This includes c-orientations (Propp), α-orientations of planar graphs (Felsner/de Mendez) planar flows (Khuller, Naor and Klein) as well as some more special instances, e.g., spanning trees of a planar graph, matchings of planar bipartite graphs and Schnyder woods. We provide a characterization of upper locally distributi...

The orientifold planar equivalence is the equivalence in the large-N limit of the bosonic sectors of the super Yang-Mills and the QCD with a quark in the antisymmetric representation. I give a sketch of the proof of the orientifold planar equivalence in the strong-coupling and large-mass phase on the lattice. It is still matter of discussion, if its validity extends also in the continuum limit.

It is widely believed that the critical properties of several planar lattice models, like the Eight Vertex or the Ashkin-Teller models, are well described by an effective Quantum Field Theory obtained as formal scaling limit. On the basis of this assumption several extended scaling relations among their indices were conjectured. We prove the validity of some of them, among which the ones by Kad...

A central problem in subfactor theory is the classification of inclusions of II1 factors, N ⊆M . An important invariant for such an inclusion is the lattice of higher relative commutants, {M ′ i ∩Mj}i,j , known as the standard invariant, contained in the Jones tower N ⊆ M ⊆ M1 · · ·. There are several approaches to studying the standard invariant, namely paragroups [3], λ-lattices [8], and plan...

By Monte Carlo simulations in the constant-temperature--constant-pressure ensemble a planar interface between unmixed A-rich and B-rich phases of a binary (A, B) alloy on a compressible diamond lattice is studied. No significant capillary wave broadening of the concentration profile across the interface is observed, unlike lattice models of incompressible mixtures and fluids. The distortion of ...

We present novel results for the three-gluon vertex, obtained from an extensive quenched lattice simulation in Landau gauge. The evaluates transversely projected spanned on a special tensorial basis, whose form factors are naturally parametrized terms of individually Bose-symmetric variables. Quite interestingly, when evaluated these kinematics, corresponding depend almost exclusively single ki...