A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods

We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme

Exponential reduction of finite volume effects with twisted boundary conditions

Flavor-twisted boundary conditions can be used for exponential reduction of finite volume artifacts in flavor-averaged observables in lattice QCD calculations with $SU(N_f)$ light quark flavor symmetry. Finite volume artifact reduction arises from destructive interference effects in a manner closely related to the phase averaging which leads to large $N_c$ volume independence. With a particular choice of flavor-twisted boundary conditions, finite volume artifacts for flavor-singlet observables in a hypercubic spacetime volume are reduced to the size of finite volume artifacts in a spacetime volume with periodic boundary conditions that is four times larger.Comment: 18 pages, no figure

Kinks in Finite Volume

A (1+1)-dimensional quantum field theory with a degenerate vacuum (in infinite volume) can contain particles, known as kinks, which interpolate between different vacua and have nontrivial restrictions on their multi-particle Hilbert space. Assuming such a theory to be integrable, we show how to calculate the multi-kink energy levels in finite volume given its factorizable $S$-matrix. In massive theories this can be done exactly up to contributions due to off-shell and tunneling effects that fall off exponentially with volume. As a first application we compare our analytical predictions for the kink scattering theories conjectured to describe the subleading thermal and magnetic perturbations of the tricritical Ising model with numerical results from the truncated conformal space approach. In particular, for the subleading magnetic perturbation our results allow us to decide between the two different $S$-matrices proposed by Smirnov and Zamolodchikov.Comment: 48/28 pages + 10 figs, 4 in pictex, the rest in postscript files attached at the en

A geometrically bounding hyperbolic link complement

A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume is roughly equal to 29.311.Comment: 23 pages, 19 figure

Quenched and Unquenched Chiral Perturbation Theory in the \epsilon-Regime

The chiral limit of finite-volume QCD is the $\epsilon$-regime of the theory. We discuss how this regime can be used for determining low-energy observables of QCD by means of comparisons between lattice simulations and quenched and unquenched chiral perturbation theory. The quenched theory suffers in the $\epsilon$-regime from ``quenched finite volume logs'', the finite-volume analogs of quenched chiral logs.Comment: LaTeX, 7 pages, contribution to LHP200

Volume Dependence of the Pion Mass from Renormalization Group Flows

We investigate finite volume effects on the pion mass and the pion decay constant with renormalization group (RG) methods in the framework of a phenomenological model for QCD. An understanding of such effects is important in order to interpret results from lattice QCD and extrapolate reliably from finite lattice volumes to infinite volume. We consider the quark-meson-model in a finite Euclidean 3+1 dimensional volume. In order to break chiral symmetry in the finite volume, we introduce a small current quark mass. In the corresponding effective potential for the meson fields, the chiral O(4)-symmetry is broken explicitly, and the sigma and pion fields are treated individually. Using the proper-time renormalization group, we derive renormalization group flow equations in the finite volume and solve these equations in the approximation of a constant expectation value. We calculate the volume dependence of pion mass and pion decay constant and compare our results with recent results from chiral perturbation theory in finite volume.Comment: 9 pages, 3 figures, talk given at "Hadronic Physics 2004 - Joint meeting Heidelberg-Liege-Paris-Rostock", to appear in the proceedings, AIP conference serie

Infrared Renormalons and Finite Volume

We analyze the perturbative expansion of a condensate in the O(N) non-linear sigma model for large N on a two dimensional finite lattice. On an infinite volume this expansion is affected by an infrared renormalon. We extrapolate this analysis to the case of the gluon condensate of Yang-Mills theory and argue that infrared renormalons can be detected by performing perturbative studies even on relatively small lattices.Comment: LaTeX file, 6 figures in postscrip

Finite Volume Spaces and Sparsification

We introduce and study finite $d$-volumes - the high dimensional generalization of finite metric spaces. Having developed a suitable combinatorial machinery, we define $\ell_1$-volumes and show that they contain Euclidean volumes and hypertree volumes. We show that they can approximate any $d$-volume with $O(n^d)$ multiplicative distortion. On the other hand, contrary to Bourgain's theorem for $d=1$, there exists a $2$-volume that on $n$ vertices that cannot be approximated by any $\ell_1$-volume with distortion smaller than $\tilde{\Omega}(n^{1/5})$. We further address the problem of $\ell_1$-dimension reduction in the context of $\ell_1$ volumes, and show that this phenomenon does occur, although not to the same striking degree as it does for Euclidean metrics and volumes. In particular, we show that any $\ell_1$ metric on $n$ points can be $(1+ \epsilon)$-approximated by a sum of $O(n/\epsilon^2)$ cut metrics, improving over the best previously known bound of $O(n \log n)$ due to Schechtman. In order to deal with dimension reduction, we extend the techniques and ideas introduced by Karger and Bencz{\'u}r, and Spielman et al.~in the context of graph Sparsification, and develop general methods with a wide range of applications.Comment: previous revision was the wrong file: the new revision: changed (extended considerably) the treatment of finite volumes (see revised abstract). Inserted new applications for the sparsification technique

Pion mass dependence of the Kl3K_{l3} semileptonic scalar form factor within finite volume

We calculate the scalar semileptonic kaon decay in finite volume at the momentum transfer $t_{m} = (m_{K} - m_{\pi})^2$, using chiral perturbation theory. At first we obtain the hadronic matrix element to be calculated in finite volume. We then evaluate the finite size effects for two volumes with $L = 1.83 fm$ and $L= 2.73 fm$ and find that the difference between the finite volume corrections of the two volumes are larger than the difference as quoted in \cite{Boyle2007a}. It appears then that the pion masses used for the scalar form factor in ChPT are large which result in large finite volume corrections. If appropriate values for pion mass are used, we believe that the finite size effects estimated in this paper can be useful for Lattice data to extrapolate at large lattice size.Comment: 19 pages, 5 figures, accepted for publication in EPJ