From Cycle Rooted Spanning Forests to the Critical Ising Model: an Explicit Construction

Fisher established an explicit correspondence between the 2-dimensional Ising model defined on a graph G and the dimer model defined on a decorated version ${\mathcal{G}}$ of this graph (Fisher in J Math Phys 7:1776–1781, 1966). In this paper we explicitly relate the dimer model associated to the critical Ising model and critical cycle rooted spanning forests (CRSFs). This relation is established through characteristic polynomials, whose definition only depends on the respective fundamental domains, and which encode the combinatorics of the model. We first show a matrix-tree type theorem establishing that the dimer characteristic polynomial counts CRSFs of the decorated fundamental domain ${\mathcal{G}_1}$ . Our main result consists in explicitly constructing CRSFs of ${\mathcal{G}_1}$ counted by the dimer characteristic polynomial, from CRSFs of G ₁, where edges are assigned Kenyon’s critical weight function (Kenyon in Invent Math 150(2):409–439, 2002); thus proving a relation on the level of configurations between two well known 2-dimensional critical models.     

Approximate Lifshitz Law for the Zero-Temperature Stochastic Ising Model in any Dimension

We study the Glauber dynamics for the zero-temperature stochastic Ising model in dimension d ≥ 4 with “plus” boundary condition. Let ${\mathcal{T}_+}$ be the time needed for an hypercube of size L entirely filled with “minus” spins to become entirely “plus”. We prove that ${\mathcal{T}_+}$ is O(L ²(log L) ^c ) for some constant c, not depending on the dimension. This brings further rigorous justification for the so-called “Lifshitz law” ${\mathcal{T}_{+} = O(L^{2})}$ (Fischer and Huse in Phys Rev B 35:6841–6848, 1987; Lifshitz in Sov Phys JETP 15:939–942, 1962) conjectured on heuristic grounds. The key point of our proof is to use the detailed knowledge that we have on the three-dimensional problem: results for fluctuation of monotone interfaces at equilibrium and mixing time for monotone interfaces dynamics extracted from Caputo et al. (Comm Pure Appl Math 64:778–831, 2011) to get the result in higher dimension.     

Composite and Elementary Components in Hadron Resonances

In hadron resonances different structures of hadronic composite (molecule) and elementary (quark-intrinsic) natures may coexist. We sketch discussions based on our previous publications on the origin of hadron resonances (Hyodo et al. Phys. Rev. C 78:025203, 2008) on exotic ${\bar D (B)}$ meson–nucleons as candidates of hadronic composites (Yamaguchi et al. Phys. Rev. D 84:014032, 2011) and on a ₁ for the coexistence/mixing of the two different natures (Nagahiro et al. Phys. Rev. D 83:111504, 2011).     

Threshold Phenomenon for the Quintic Wave Equation in Three Dimensions

For the critical focusing wave equation ${\square u = u^5 \, {\rm on} \, \mathbb{R}^{3+1}}$ in the radial case, we establish the role of the “center stable” manifold ${\Sigma}$ constructed in Krieger and Schlag (Am J Math 129(3):843–913, 2007) near the ground state (W, 0) as a threshold between blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizoń et al. (Nonlinearity 17(6):2187–2201, 2004). The underlying topology is stronger than the energy norm.     

Semiclassical Approximations for Hamiltonians with Operator-Valued Symbols

We consider the semiclassical limit of quantum systems with a Hamiltonian given by the Weyl quantization of an operator valued symbol. Systems composed of slow and fast degrees of freedom are of this form. Typically a small dimensionless parameter ${\varepsilon \ll 1}$ controls the separation of time scales and the limit ${\varepsilon\to 0}$ corresponds to an adiabatic limit, in which the slow and fast degrees of freedom decouple. At the same time ${\varepsilon\to 0}$ is the semiclassical limit for the slow degrees of freedom. In this paper we show that the ${\varepsilon}$ -dependent classical flow for the slow degrees of freedom first discovered by Littlejohn and Flynn (Phys Rev A (3) 44(8):5239–5256, 1991), coming from an ${\varepsilon}$ -dependent classical Hamilton function and an ${\varepsilon}$ -dependent symplectic form, has a concrete mathematical and physical meaning: Based on this flow we prove a formula for equilibrium expectations, an Egorov theorem and transport of Wigner functions, thereby approximating properties of the quantum system up to errors of order ${\varepsilon^2}$ . In the context of Bloch electrons formal use of this classical system has triggered considerable progress in solid state physics (Xiao et al. in Rev Mod Phys 82(3):1959–2007, 2010). Hence we discuss in some detail the application of the general results to the Hofstadter model, which describes a two-dimensional gas of non-interacting electrons in a constant magnetic field in the tight-binding approximation.     

Testing the Concept of Quark–Hadron Duality with the ALEPH τ Decay Data

We propose a modified procedure for extracting the numerical value for the strong coupling constant α _s from the τ lepton hadronic decay rate into non-strange particles in the vector channel. We employ the concept of the quark–hadron duality specifically, introducing a boundary energy squared s _p > 0, the onset of the perturbative QCD continuum in Minkowski space (Bertlmann et al. in Nucl Phys B 250:61, 1985; de Rafael in An introduction to sum rules in QCD. In: Lectures at the Les Houches Summer School. arXiv: 9802448 [hep-ph], 1997; Peris et al. in JHEP 9805:011, 1998). To approximate the hadronic spectral function in the region s > s _p, we use analytic perturbation theory (APT) up to the fifth order. A new feature of our procedure is that it enables us to extract from the data simultaneously the QCD scale parameter ${\Lambda_{\overline{\rm MS}}}$ and the boundary energy squared s _p. We carefully determine the experimental errors on these parameters which come from the errors on the invariant mass squared distribution. For the ${\overline{\rm MS}}$ scheme coupling constant, we obtain ${\alpha_s(m^{2}_{\tau})=0.3204\pm 0.0159_{exp.}}$ . We show that our numerical analysis is much more stable against higher-order corrections than the standard one. Additionally, we recalculate the “experimental” Adler function in the infrared region using final ALEPH results. The uncertainty on this function is also determined.     

Predicting leptonic CP violation in the light of the Daya Bay result on θ 13

In the light of the recent Daya Bay result $\theta_{13}^{\mathrm{DB}}=8.8^{\circ}\pm0.8^{\circ}$ , we reconsider the model presented in Meloni et?al. (J. Phys.?G 38:015003, 2011), showing that, when all neutrino oscillation parameters are taken at their best fit values of Schwetz et?al. (New J. Phys. 10:113011,?2008) and where $\theta_{13}=\theta_{13}^{\mathrm{DB}}$ , the predicted values of the CP phase are ????±??/4.     

A note on the area spectrum of d-dimensional pure de Sitter Space-time

Consistent supercurrent multiplets are naturally associated with linearized off-shell supergravity models. In S.M. Kuzenko, J. High Energy Phys. 1004, 022 (2010) we presented the hierarchy of such supercurrents which correspond to all the models for linearized 4D $\mathcal{N}=1$ supergravity classified a few years ago. Here we analyze the correspondence between the most general supercurrent given in S.M. Kuzenko, J. High Energy Phys. 1004, 022 (2010) and the one obtained eight years ago in M. Magro et al., Ann. Phys. 298, 123 (2002) using the superfield Noether procedure. We apply the Noether procedure to the general $\mathcal{N}=1$ supersymmetric nonlinear sigma-model and show that it naturally leads to the so-called $\mathcal{S}$ -multiplet, revitalized in Z. Komargodski, N. Seiberg, J. High Energy Phys. 1007, 017 (2010).     

Open Gromov-Witten Invariants of Toric Calabi-Yau 3-Folds

We present a proof of the mirror conjecture of Aganagic and Vafa (Mirror Symmetry, D-Branes and Counting Holomorphic Discs. http://arxiv.org/abs/hep-th/0012041v1, 2000) and Aganagic et al. (Z Naturforsch A 57(1–2):128, 2002) on disk enumeration in toric Calabi-Yau 3-folds for all smooth semi-projective toric Calabi-Yau 3-folds. We consider both inner and outer branes, at arbitrary framing. In particular, we recover previous results on the conjecture for (i) an inner brane at zero framing in ${K_{\mathbb{P}^2}}$ K P 2 (Graber-Zaslow, Contemp Math 310:107–121, 2002), (ii) an outer brane at arbitrary framing in the resolved conifold ${\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)}$ O P 1 ( - 1 ) ⊕ O P 1 ( - 1 ) (Zhou, Open string invariants and mirror curve of the resolved conifold. http://arxiv.org/abs/1001.0447v1 [math.AG], 2010), and (iii) an outer brane at zero framing in ${K_{\mathbb{P}^2}}$ K P 2 (Brini, Open topological strings and integrable hierarchies: Remodeling the A-model. http://arxiv.org/abs/1102.0281 [hep-th], 2011).     

Accurate Expansions of Internal Energy and Specific Heat of Critical Two-Dimensional Ising Model with Free Boundaries

The bond-propagation algorithm for the specific heat of the two dimensional Ising model is developed and that for the internal energy is completed. Using these algorithms, we study the critical internal energy and specific heat of the model on the square lattice and triangular lattice with free boundaries. Comparing with previous works (Phys Rev E 86:041149, 2012; Phys Rev E 87:022124, 2013), we reach much higher accuracy ( $10^{-28}$ ) of the internal energy and specific heat, compared to the accuracy $10^{-11}$ of the internal energy and $10^{-9}$ of the specific heat reached in the previous works. This leads to much more accurate estimations of the edge and corner terms. The exact values of all edge and corner terms are therefore conjectured. The accurate forms of finite-size scaling for the internal energy and specific heat are determined for the rectangle-shaped square lattice with various aspect ratios and various shaped triangular lattice. For the rectangle-shaped square and triangular lattices and the triangle-shaped triangular lattice, there is no logarithmic correction terms of order higher than $1/S$ , with $S$ the area of the system. For the triangular lattice in rhombus, trapezoid and hexagonal shapes, there exist logarithmic correction terms of order higher than $1/S$ for the internal energy, and logarithmic correction terms of all orders for the specific heat.     

Non-Equilibrium Statistical Mechanics of Turbulence

The macroscopic study of hydrodynamic turbulence is equivalent, at an abstract level, to the microscopic study of a heat flow for a suitable mechanical system (Ruelle, PNAS 109:20344–20346, 2012). Turbulent fluctuations (intermittency) then correspond to thermal fluctuations, and this allows to estimate the exponents \(\tau _p\) and \(\zeta _p\) associated with moments of dissipation fluctuations and velocity fluctuations. This approach, initiated in an earlier note (Ruelle, 2012), is pursued here more carefully. In particular we derive probability distributions at finite Reynolds number for the dissipation and velocity fluctuations, and the latter permit an interpretation of numerical experiments (Schumacher, Preprint, 2014). Specifically, if \(p(z)dz\) is the probability distribution of the radial velocity gradient we can explain why, when the Reynolds number \(\mathcal{R}\) increases, \(\ln p(z)\) passes from a concave to a linear then to a convex profile for large \(z\) as observed in (Schumacher, 2014). We show that the central limit theorem applies to the dissipation and velocity distribution functions, so that a logical relation with the lognormal theory of Kolmogorov (J. Fluid Mech. 13:82–85, 1962) and Obukhov is established. We find however that the lognormal behavior of the distribution functions fails at large value of the argument, so that a lognormal theory cannot correctly predict the exponents \(\tau _p\) and \(\zeta _p\) .     

Frame-dragging fields and spin 1 gravitomagnetic radiation

Experimental results published in 2004 (Ciufolini and Pavlis in Nature 431:958–960, 2004) and 2011 (Everitt et al. in Phys Rev Lett 106:221101, 1–5, 2011) have confirmed the frame-dragging phenomenon for a spinning earth predicted by Einstein’s field equations. Since this is observed as a precession caused by the gravitomagnetic (GM) field of the rotating body, these experiments may be viewed as measurements of a GM field. The effect is encapsulated in the classic steady state solution for the vector potential field $\zeta $ of a spinning sphere–a solution applying to a sphere with angular momentum J and describing a field filling space for all time (Weinberg in Gravitation and Cosmology, Wiley, New York, 1972). In a laboratory setting one may visualise the case of a sphere at rest $(\zeta =0, \text{ t}<0)$ , being spun up by an external torque at $\text{ t}=0$ to the angular momentum J: the $\zeta $ field of the textbook solution cannot establish itself instantaneously over all space at $\text{ t}=0$ , but must propagate with the velocity c, implying the existence of a travelling GM wave field yielding the textbook $\zeta $ field for large enough t (Tolstoy in Int J Theor Phys 40(5):1021–1031, 2001). The linearized GM field equations of the post-Newtonian approximation being isomorphic with Maxwell’s equations (Braginsky et al. in Phys Rev D 15(6):2047–2060, 1977), such GM waves are dipole waves of spin 1. It is well known that in purely gravitating systems conservation of angular momentum forbids the existence of dipole radiation (Misner et al. in Gravitation, Freeman & Co., New York, 1997); but this rule does not prohibit the insertion of angular momentum into the system from an external source–e.g., by applying a torque to our laboratory sphere.     

Extensions of Ordering Sets of States from Effect Algebras onto Their MacNeille Completions

In (Rie?anová and Zajac in Rep. Math. Phys. 70(2):283–290, 2012) it was shown that an effect algebra E with an ordering set $\mathcal{M}$ of states can by embedded into a Hilbert space effect algebra $\mathcal{E}(l_{2}(\mathcal{M}))$ . We consider the problem when its effect algebraic MacNeille completion $\hat{E}$ can be also embedded into the same Hilbert space effect algebra $\mathcal {E}(l_{2}(\mathcal{M}))$ . That is when the ordering set $\mathcal{M}$ of states on E can be extended to an ordering set of states on $\hat{E}$ . We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.     

Sharp parameter bounds for certain maximal point lenses

Starting from an $n$ -point circular gravitational lens having $3n+1$ images, Rhie (ArXiv Astrophysics e-prints, 2003) used a perturbation argument to construct an $(n+1)$ -point lens producing $5n$ images. In this work we give a concise proof of Rhie’s result, and we extend the range of parameters in Rhie’s model for which maximal lensing occurs. We also study a slightly different construction given by Bayer and Dyer (Gen Relativ Gravit 39(9):1413–1418, 2007) arising from the $(3n+1)$ -point lens. In particular, we extend their results and give sharp parameter bounds for their lens model. By a substitution of variables and parameters we show that both models are equivalent in a certain sense.     

Quantum Curves for Hitchin Fibrations and the Eynard–Orantin Theory

We generalize the topological recursion of Eynard–Orantin (JHEP 0612:053, 2006; Commun Number Theory Phys 1:347–452, 2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle T*C of an arbitrary smooth base curve C. We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal ${\hbar}$ -deformation family of D modules over an arbitrary projective algebraic curve C of genus greater than 1, from the geometry of a prescribed family of smooth Hitchin spectral curves associated with the ${SL(2,\mathbb{C})}$ -character variety of the fundamental group π₁(C). We show that the semi-classical limit through the WKB approximation of these ${\hbar}$ -deformed D modules recovers the initial family of Hitchin spectral curves.     

Addendum to: A new numerical method for obtaining gluon distribution functions G(x,Q 2)=xg(x,Q 2), from the proton structure function F_{2}^{\gamma p}(x,Q^{2})

Since publication of M.M. Block in Eur. Phys. J. C 65, 1 (2010), we have discovered that the algorithm of Block (2010) does not work if g(s)→0 less rapidly than 1/s, as s→∞. Although we require that g(s)→0 as s→∞, it can approach 0 as ${1\over s^{\beta}}$ , with 0<β<1, and still be a proper Laplace transform. In this note, we derive a new numerical algorithm for just such cases, and test it for $g(s)={\sqrt{\pi}\over \sqrt{s}}$ , the Laplace transform of ${1\over\sqrt{v}}$ .     

Tau-Function Theory of Chaotic Quantum Transport with β = 1, 2, 4

We study the cumulants and their generating functions of the probability distributions of the conductance, shot noise and Wigner delay time in ballistic quantum dots. Our approach is based on the integrable theory of certain matrix integrals and applies to all the symmetry classes ${\beta \in \{1, 2, 4\}}$ of Random Matrix Theory. We compute the weak localization corrections to the mixed cumulants of the conductance and shot noise for β = 1, 4, thus proving a number of conjectures of Khoruzhenko et al. (in Phys Rev B 80:(12)125301, 2009). We derive differential equations that characterize the cumulant generating functions for all ${\beta \in \{1, 2, 4 \} }$ . Furthermore, when β = 2 we show that the cumulant generating function of the Wigner delay time can be expressed in terms of the Painlevé III′ transcendant. This allows us to study properties of the cumulants of the Wigner delay time in the asymptotic limit ${n \to \infty}$ . Finally, for all the symmetry classes and for any number of open channels, we derive a set of recurrence relations that are very efficient for computing cumulants at all orders.     

Antiproton reflection by a solid surface

The AE?IS experiment (Antimatter Experiment: Gravity, Interferometry, Spectroscopy (Drobychev et al., 2007)), aims at directly measuring the gravitational acceleration g on a beam of cold antihydrogen ( $\overline{\rm H}$ ). After production, the $\overline{\rm H}$ atoms will be driven to fly horizontally with a velocity of a few 100 m/s for a path length of about 1 meter. The small deflection, few tens of μm, will be measured using two material gratings coupled to a position-sensitive detector working as a Moiré deflectometer similarly to what has been done with atoms (Oberthaler et al., Phys Rev A 54:3165, 1996). Details about the detection of the $\overline{\rm H}$ annihilation point at the end of the flight path with a position-sensitive microstrip detector and a silicon tracker system will be discussed.