Pre-logarithmic and logarithmic fields in a sandpile model

We consider the unoriented two-dimensional Abelian sandpile model on the half-plane with open and closed boundary conditions, and relate it to the boundary logarithmic conformal field theory with central charge c=-2. Building on previous results, we first perform a complementary lattice analysis of the operator effecting the change of boundary condition between open and closed, which confirms that this operator is a weight -1/8 boundary primary field, whose fusion agrees with lattice calculations. We then consider the operators corresponding to the unit height variable and to a mass insertion at an isolated site of the upper half plane and compute their one-point functions in presence of a boundary containing the two kinds of boundary conditions. We show that the scaling limit of the mass insertion operator is a weight zero logarithmic field.Comment: 18 pages, 9 figures. v2: minor corrections + added appendi

Logarithmic correction in the deformed AdS5{\rm AdS}_5 model to produce the heavy quark potential and QCD beta function

We stude the \textit{holographic} QCD model which contains a quadratic term $-\sigma z^2$ and a logarithmic term $-c_0\log[(z_{IR}-z)/z_{IR}]$ with an explicit infrared cut-off $z_{IR}$ in the deformed ${\rm AdS}_5$ warp factor. We investigate the heavy quark potential for three cases, i.e, with only quadratic correction, with both quadratic and logarithmic corrections and with only logarithmic correction. We solve the dilaton field and dilation potential from the Einstein equation, and investigate the corresponding beta function in the G{\"u}rsoy -Kiritsis-Nitti (GKN) framework. Our studies show that in the case with only quadratic correction, a negative $\sigma$ or the Andreev-Zakharov model is favored to fit the heavy quark potential and to produce the QCD beta-function at 2-loop level, however, the dilaton potential is unbounded in infrared regime. One interesting observing for the case of positive $\sigma$, or the soft-wall ${\rm AdS}_5$ model is that the corresponding beta-function exists an infrared fixed point. In the case with only logarithmic correction, the heavy quark Cornell potential can be fitted very well, the corresponding beta-function agrees with the QCD beta-function at 2-loop level reasonably well, and the dilaton potential is bounded from below in infrared. At the end, we propose a more compact model which has only logarithmic correction in the deformed warp factor and has less free parameters.Comment: 24 pages, 16 figure

The Kosterlitz-Thouless Universality Class

We examine the Kosterlitz-Thouless universality class and show that essential scaling at this type of phase transition is not self-consistent unless multiplicative logarithmic corrections are included. In the case of specific heat these logarithmic corrections are identified analytically. To identify those corresponding to the susceptibility we set up a numerical method involving the finite-size scaling of Lee-Yang zeroes. We also study the density of zeroes and introduce a new concept called index scaling. We apply the method to the XY-model and the closely related step model in two dimensions. The critical parameters (including logarithmic corrections) of the step model are compatable with those of the XY-model indicating that both models belong to the same universality class. This result then raises questions over how a vortex binding scenario can be the driving mechanism for the phase transition. Furthermore, the logarithmic corrections identified numerically by our methods of fitting are not in agreement with the renormalization group predictions of Kosterlitz and Thouless.Comment: 36 pages (latex), plus 10 figures (postscript). This version to appear in Nuclear Physics

The parameterized space complexity of model-checking bounded variable first-order logic

The parameterized model-checking problem for a class of first-order sentences (queries) asks to decide whether a given sentence from the class holds true in a given relational structure (database); the parameter is the length of the sentence. We study the parameterized space complexity of the model-checking problem for queries with a bounded number of variables. For each bound on the quantifier alternation rank the problem becomes complete for the corresponding level of what we call the tree hierarchy, a hierarchy of parameterized complexity classes defined via space bounded alternating machines between parameterized logarithmic space and fixed-parameter tractable time. We observe that a parameterized logarithmic space model-checker for existential bounded variable queries would allow to improve Savitch's classical simulation of nondeterministic logarithmic space in deterministic space $O(\log^2n)$. Further, we define a highly space efficient model-checker for queries with a bounded number of variables and bounded quantifier alternation rank. We study its optimality under the assumption that Savitch's Theorem is optimal

Logarithmic corrected Polynomial f(R)f(R) inflation mimicking a cosmological constant

In this paper, we consider an inflationary model of $f(R)$ gravity with polynomial form plus logarithmic term. We calculate some cosmological parameters and compare our results with the Plank 2015 data. We find that presence of both logarithmic and polynomial corrections are necessary to yield slow-roll condition. Also, we study critical points and stability of the model to find that it is a viable model.Comment: 15 pages, 9 figure

Logarithmic conformal field theories with continuous weights

We study the logarithmic conformal field theories in which conformal weights are continuous subset of real numbers. A general relation between the correlators consisting of logarithmic fields and those consisting of ordinary conformal fields is investigated. As an example the correlators of the Coulomb-gas model are explicitly studied.Comment: Latex, 12 pages, IPM preprint, to appear in Phys. Lett.

Growth models on the Bethe lattice

I report on an extensive numerical investigation of various discrete growth models describing equilibrium and nonequilibrium interfaces on a substrate of a finite Bethe lattice. An unusual logarithmic scaling behavior is observed for the nonequilibrium models describing the scaling structure of the infinite dimensional limit of the models in the Kardar-Parisi-Zhang (KPZ) class. This gives rise to the classification of different growing processes on the Bethe lattice in terms of logarithmic scaling exponents which depend on both the model and the coordination number of the underlying lattice. The equilibrium growth model also exhibits a logarithmic temporal scaling but with an ordinary power law scaling behavior with respect to the appropriately defined lattice size. The results may imply that no finite upper critical dimension exists for the KPZ equation.Comment: 5 pages, 5 figure

Scaling in the vicinity of the four-state Potts fixed point

We study a self-dual generalization of the Baxter-Wu model, employing results obtained by transfer matrix calculations of the magnetic scaling dimension and the free energy. While the pure critical Baxter-Wu model displays the critical behavior of the four-state Potts fixed point in two dimensions, in the sense that logarithmic corrections are absent, the introduction of different couplings in the up- and down triangles moves the model away from this fixed point, so that logarithmic corrections appear. Real couplings move the model into the first-order range, away from the behavior displayed by the nearest-neighbor, four-state Potts model. We also use complex couplings, which bring the model in the opposite direction characterized by the same type of logarithmic corrections as present in the four-state Potts model. Our finite-size analysis confirms in detail the existing renormalization theory describing the immediate vicinity of the four-state Potts fixed point.Comment: 19 pages, 7 figure