Abstract: This is an expository account of Balaban’s approach to the renormalization group. The method is illustrated with a treatment of the ultraviolet problem for the scalar
\({\phi^4}\)
model on a toroidal lattice in dimension d = 3. In this third paper we demonstrate convergence of the expansion and complete the proof of a stability bound. PubDate: 2014-11-01

Abstract: The vector space
\({\otimes^{n}\mathbb{C}^2}\)
upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra
\({{\rm TL}_{n}(\beta = q + q^{-1})}\)
and the quantum algebra
\({{\rm U}_{q} \mathfrak{sl}_2}\)
. The decomposition of
\({\otimes^{n}\mathbb{C}^2}\)
as a
\({{\rm U}_{q} \mathfrak{sl}_2}\)
-module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL
n
-module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL
n
-module
\({\otimes^{n}\mathbb{C}^2}\)
is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of
\({\otimes^{n}\mathbb{C}^2}\)
) onto each of these irreducible modules as linear combinations of elements of
\({{\rm U}_{q} \mathfrak{sl}_2}\)
. When q = q
c
is a root of unity, the TL
n
-module
\({\otimes^{n}\mathbb{C}^2}\)
(with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on
\({\otimes^{n}\mathbb{C}^2}\)
is that of the divided powers
\({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\)
. PubDate: 2014-11-01

Abstract: Melonic graphs constitute the family of graphs arising at leading order in the 1/N expansion of tensor models. They were shown to lead to a continuum phase, reminiscent of branched polymers. We show here that they are in fact precisely branched polymers, that is, they possess Hausdorff dimension 2 and spectral dimension 4/3. PubDate: 2014-11-01

Abstract: In the early 1980s Yau posed the problem of establishing the rigidity of the Hawking–Penrose singularity theorems. Approaches to this problem have involved the introduction of Lorentzian Busemann functions and the study of the geometry of their level sets—the horospheres. The regularity theory in the Lorentzian case is considerably more complicated and less complete than in the Riemannian case. In this paper, we introduce a broad generalization of the notion of horosphere in Lorentzian geometry and take a completely different (and highly geometric) approach to regularity. These generalized horospheres are defined in terms of achronal limits, and the improved regularity we obtain is based on regularity properties of achronal boundaries. We establish a splitting result for generalized horospheres, which when specialized to Cauchy horospheres yields new results on the Bartnik splitting conjecture, a concrete realization of the problem posed by Yau. Our methods are also applied to spacetimes with positive cosmological constant. We obtain a rigid singularity result for future asymptotically de Sitter spacetimes related to results in Andersson and Galloway (Adv Theor Math Phys 6:307–327, 2002), and Cai and Galloway (Adv Theor Math Phys 3:1769–1783, 2000). PubDate: 2014-11-01

Abstract: In this paper, we present a straightforward pictorial representation of the double affine Hecke algebra (DAHA) which enables us to translate the abstract algebraic structure of a DAHA into an intuitive graphical calculus suitable for a physics audience. Initially, we define the larger double affine Q-dependent braid group. This group is constructed by appending to the braid group a set of operators Q
i
, before extending it to an affine Q-dependent braid group. We show specifically that the elliptic braid group and the DAHA can be obtained as quotient groups. Complementing this, we present a pictorial representation of the double affine Q-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation, we can fully describe any DAHA. Specifically, we graphically describe the parameter q upon which this algebra is dependent and show that in this particular representation q corresponds to a twist in the ribbon. PubDate: 2014-11-01

Abstract: In this paper, we reformulate the combinatorial core of constructive quantum field theory. We define universal rational combinatorial weights for pairs made of a graph and any of its spanning trees. These weights are simply the percentage of Hepp’s sectors of the graph in which the tree is leading, in the sense of Kruskal’s greedy algorithm. Our main new mathematical result is an integral representation of these weights in terms of the positive matrix appearing in the symmetric “BKAR” Taylor forest formula. Then, we explain how the new constructive technique called Loop Vertex Expansion reshuffles according to these weights the divergent series of the intermediate field representation into a convergent series which is the Borel sum of the ordinary perturbative Feynman’s series. PubDate: 2014-11-01

Abstract: Abstract
We consider general cyclic representations of the six-vertex Yang–Baxter algebra and analyze the associated quantum integrable systems, the Bazhanov–Stroganov model and the corresponding chiral Potts model on finite size lattices. We first determine the propagator operator in terms of the chiral Potts transfer matrices and we compute the scalar product of separate states (including the transfer matrix eigenstates) as a single determinant formulae in the framework of Sklyanin’s quantum separation of variables. Then, we solve the quantum inverse problem and reconstruct the local operators in terms of the separate variables. We also determine a basis of operators whose form factors are characterized by a single determinant formulae. This implies that the form factors of any local operator are expressed as finite sums of determinants. Among these form factors written in determinant form are in particular those which will reproduce the chiral Potts order parameters in the thermodynamic limit. The results presented here are the generalization to the present models associated to the most general cyclic representations of the six-vertex Yang–Baxter algebra of those we derived for the lattice sine–Gordon model. PubDate: 2014-10-22

Abstract: Abstract
Random operators may acquire extended states formed from a multitude of mutually resonating local quasi-modes. This mechanics is explored here in the context of the random Schrödinger operator on the complete graph. The operator exhibits local quasi-modes mixed through a single channel. While most of its spectrum consists of localized eigenfunctions, under appropriate conditions it includes also bands of states which are delocalized in the
\({\ell^{1}}\)
-though not in
\({\ell^{2}}\)
-sense, where the eigenvalues have the statistics of Šeba spectra. The analysis proceeds through some general observations on the scaling limits of random functions in the Herglotz–Pick class. The results are in agreement with a heuristic condition for the emergence of resonant delocalization, which is stated in terms of the tunneling amplitude among quasi-modes. PubDate: 2014-10-18

Abstract: Abstract
It is shown that solutions to Einstein’s field equations with positive cosmological constant can include non-zero rest-mass fields which coexist with and travel unimpeded across a smooth conformal boundary. This is exemplified by the coupled Einstein-massive-scalar field equations for which the mass m is related to the cosmological constant λ by the relation 3m
2 = 2 λ. Cauchy data for the conformal field equations can in this case be prescribed on the (compact, space-like) conformal boundary
\({\mathcal{J}^{+}}\)
. Their developments backwards in time induce a set of standard Cauchy data on space-like slices for the Einstein-massive-scalar field equations which is open in the set of all Cauchy data for this system. PubDate: 2014-10-18

Abstract: Abstract
For random operators it is conjectured that spectral properties of an infinite-volume operator are related to the distribution of spectral gaps of finite-volume approximations. In particular, localization and pure point spectrum in infinite volume is expected to correspond to Poisson eigenvalue statistics. Motivated by results on the Anderson model on the infinite tree we consider random Schrödinger operators on finite regular graphs. We study local spectral statistics: we analyze the number of eigenvalues in intervals with length comparable to the inverse of the number of vertices of the graph, in the limit where this number tends to infinity. We show that the random point process generated by the rescaled eigenvalues converges in certain spectral regimes of localization to a Poisson process. The corresponding result on the lattice was proved by Minami. However, due to the geometric structure of regular graphs the known methods turn out to be difficult to adapt. Therefore, we develop a new approach based on direct comparison of eigenvectors. PubDate: 2014-10-16

Abstract: Abstract
We provide a detailed analysis of the classical and quantized theory of a multiplet of inhomogeneous Klein–Gordon fields, which couple to the spacetime metric and also to an external source term; thus the solutions form an affine space. Following the formulation of affine field theories in terms of presymplectic vector spaces as proposed in Benini et al. (Ann. Henri Poincaré 15:171–211, 2014), we determine the relative Cauchy evolution induced by metric as well as source term perturbations and compute the automorphism group of natural isomorphisms of the presymplectic vector space functor. Two pathological features of this formulation are revealed: the automorphism group contains elements that cannot be interpreted as global gauge transformations of the theory; moreover, the presymplectic formulation does not respect a natural requirement on composition of subsystems. We therefore propose a systematic strategy to improve the original description of affine field theories at the classical and quantized level, first passing to a Poisson algebra description in the classical case. The idea is to consider state spaces on the classical and quantum algebras suggested by the physics of the theory (in the classical case, we use the affine solution space). The state spaces are not separating for the algebras, indicating a redundancy in the description. Removing this redundancy by a quotient, a functorial theory is obtained that is free of the above-mentioned pathologies. These techniques are applicable to general affine field theories and Abelian gauge theories. The resulting quantized theory is shown to be dynamically local. PubDate: 2014-10-16

Abstract: We consider the question whether a static potential on an asymptotically flat 3-manifold can have nonempty zero set which extends to the infinity. We prove that this does not occur if the metric is asymptotically Schwarzschild with nonzero mass. If the asymptotic assumption is relaxed to the usual assumption under which the total mass is defined, we prove that the static potential is unique up to scaling unless the manifold is flat. We also provide some discussion concerning the rigidity of complete asymptotically flat 3-manifolds without boundary that admit a static potential. PubDate: 2014-10-15

Abstract: We prove that compact Cauchy horizons in a smooth spacetime satisfying the null energy condition are smooth. As an application, we consider the problem of determining when a cobordism admits Lorentzian metrics with certain properties. In particular, we prove a result originally due to Tipler without the smoothness hypothesis necessary in the original proof. PubDate: 2014-10-15

Abstract: We study the existence and uniqueness of solutions to the static vacuum Einstein equations in bounded domains, satisfying the Bartnik boundary conditions of prescribed metric and mean curvature on the boundary. PubDate: 2014-10-14

Abstract: We consider the problem of conditioning a Markov process on a rare event and of representing this conditioned process by a conditioning-free process, called the effective or driven process. The basic assumption is that the rare event used in the conditioning is a large deviation-type event, characterized by a convex rate function. Under this assumption, we construct the driven process via a generalization of Doob’s h-transform, used in the context of bridge processes, and show that this process is equivalent to the conditioned process in the long-time limit. The notion of equivalence that we consider is based on the logarithmic equivalence of path measures and implies that the two processes have the same typical states. In constructing the driven process, we also prove equivalence with the so-called exponential tilting of the Markov process, often used with importance sampling to simulate rare events and giving rise, from the point of view of statistical mechanics, to a nonequilibrium version of the canonical ensemble. Other links between our results and the topics of bridge processes, quasi-stationary distributions, stochastic control, and conditional limit theorems are mentioned. PubDate: 2014-10-12

Abstract: We study general quantum waveguides and establish explicit effective Hamiltonians for the Laplacian on these spaces. A conventional quantum waveguide is an
\({\epsilon}\)
-tubular neighbourhood of a curve in
\({\mathbb{R}^3}\)
and the object of interest is the Dirichlet Laplacian on this tube in the asymptotic limit
\({\epsilon\ll1}\)
. We generalise this by considering fibre bundles M over a complete d-dimensional submanifold
\({B\subset\mathbb{R}^{d+k}}\)
with fibres diffeomorphic to
\({F\subset\mathbb{R}^k}\)
, whose total space is embedded into an
\({\epsilon}\)
-neighbourhood of B. From this point of view, B takes the role of the curve and F that of the disc-shaped cross section of a conventional quantum waveguide. Our approach allows, among other things, for waveguides whose cross sections F are deformed along B and also the study of the Laplacian on the boundaries of such waveguides. By applying recent results on the adiabatic limit of Schrödinger operators on fibre bundles we show, in particular, that for small energies the dynamics and the spectrum of the Laplacian on M are reflected by the adiabatic approximation associated with the ground state band of the normal Laplacian. We give explicit formulas for the accordingly effective operator on L
2(B) in various scenarios, thereby improving and extending many of the known results on quantum waveguides and quantum layers in
\({\mathbb{R}^3}\)
. PubDate: 2014-10-12

Abstract: The loop vertex expansion (LVE) is a constructive technique which uses only canonical combinatorial tools and no space–time dependent lattices. It works for quantum field theories without renormalization. Renormalization requires scale analysis. In this paper, we provide an enlarged formalism which we call the multiscale loop vertex expansion (MLVE). We test it on what is probably the simplest quantum field theory which requires some kind of renormalization, namely a combinatorial model of the vector type with quartic interaction and a propagator which mimicks the power counting of
\({\phi^4_2}\)
. An ordinary LVE would fail to treat even this simplest superrenormalizable model, but we show how to perform the ultraviolet limit and prove its analyticity in the Borel summability domain of the model with the MLVE. PubDate: 2014-10-11

Abstract: We prove the spectral instability of the complex cubic oscillator
\({-\frac{{\rm d}^{2}}{{\rm d}x^{2}} + ix^{3} + i \alpha x}\)
for non-negative values of the parameter α, by getting the exponential growth rate of
\({\ \Pi_{n}(\alpha)\ }\)
, where
\({\Pi_{n}(\alpha)}\)
is the spectral projection associated with the nth eigenvalue of the operator. More precisely, we show that for all non-negative α
$$\lim\limits_{n \to + \infty} \frac{1}{n} {\rm log}\ \Pi_{n}(\alpha)\ = \frac{\pi}{\sqrt{3}}$$
. PubDate: 2014-10-01

Abstract: The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces
\({{\bf \mathcal{S}}}\)
in the Minkowski spacetime, subject to an additional convexity assumption. In a recent paper, Brendle and Wang A (Gibbons–Penrose inequality for surfaces in Schwarzschild Spacetime. arXiv:1303.1863, 2013) find a sufficient condition for the validity of this Penrose inequality in terms of the geometry of the orthogonal projection of
\({{\bf \mathcal{S}}}\)
onto a constant time hyperplane. In this work, we study the geometry of hypersurfaces in n-dimensional Euclidean space which are normal graphs over other surfaces and relate the intrinsic and extrinsic geometry of the graph with that of the base hypersurface. These results are used to rewrite Brendle and Wang’s condition explicitly in terms of the time height function of
\({{\bf \mathcal{S}}}\)
over a hyperplane and the geometry of the projection of
\({{\bf \mathcal{S}}}\)
along its past null cone onto this hyperplane. We also include, in Appendix, a self-contained summary of known and new results on the geometry of projections along the Killing direction of codimension two-spacelike surfaces in a strictly static spacetime. PubDate: 2014-10-01

Abstract: We show how to relate the full quantum dynamics of a spin-½ particle on
\({\mathbb{R}^d}\)
to a classical Hamiltonian dynamics on the enlarged phase space
\({\mathbb{R}^{2d} \times \mathbb{S}^{2}}\)
up to errors of second order in the semiclassical parameter. This is done via an Egorov-type theorem for normal Wigner–Weyl calculus for
\({\mathbb{R}^d}\)
(Folland, Harmonic Analysis on Phase Space, 1989; Lein, Weyl Quantization and Semiclassics, 2010) combined with the Stratonovich–Weyl calculus for SU(2) (Varilly and Gracia-Bondia, Ann Phys 190:107–148, 1989). For a specific class of Hamiltonians, including the Rabi- and Jaynes–Cummings model, we prove an Egorov theorem for times much longer than the semiclassical time scale. We illustrate the approach for a simple model of the Stern–Gerlach experiment. PubDate: 2014-10-01