Abstract: Abstract In this paper, we prove two existence results of solutions to mean field equations $$\begin{aligned} \Delta u+e^u=\rho \delta _0 \end{aligned}$$and $$\begin{aligned} \Delta u=\lambda e^u(e^u-1)+4 \pi \sum _{j=1}^{M}{\delta _{p_j}} \end{aligned}$$on an arbitrary connected finite graph, where \(\rho >0\) and \(\lambda >0\) are constants, M is a positive integer, and \(p_1,\ldots ,p_M\) are arbitrarily chosen distinct vertices on the graph. PubDate: 2020-02-20

Abstract: Abstract The degeneration of the hyperelliptic sigma function is studied. We use the Sato Grassmannian for this purpose. A simple decomposition of a rational function gives a decomposition of Plücker coordinates of a frame of the Sato Grassmannian. It then gives a decomposition of the tau function corresponding to the degeneration of a hyperelliptic curve of genus g in terms of the tau functions corresponding to a hyperelliptic curve of genus \(g-1\). Since the tau functions are described by sigma functions, we get the corresponding formula for the degenerate hyperelliptic sigma function. PubDate: 2020-02-19

Abstract: Abstract Consider the Cauchy problem of incompressible Navier–Stokes equations in \(\mathbb {R}^3\) with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the existence of a time-global weak solution has been known. However, such data do not include constants, and the only known global solutions for non-decaying data are either for perturbations of constants, or when the velocity gradients are in \(L^p\) with finite p. In this paper, we construct global weak solutions for non-decaying initial data whose local oscillations decay, no matter how slowly. PubDate: 2020-02-19

Abstract: Abstract This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. We give the complete classification of such curves: conformal Markov property and symmetry single out a two-parameter family of random curves—Hypergeometric SLE—denoted by \(\mathrm {hSLE}_{\kappa }(\nu )\) for \(\kappa \in (0,4]\) and \(\nu <\kappa -6\). The proof relies crucially on Dubédat’s commutation relation (Commun Pure Appl Math 60(12):1792–1847, 2007) and a uniqueness result proved in Miller and Sheffield (Ann Probab 44(3):1647–1722, 2016). The classification indicates that hypergeometric SLE is the only possible scaling limit of the interfaces in critical lattice models (conjectured or proved to be conformally invariant) in topological rectangles with alternating boundary conditions. We also prove various properties of \(\mathrm {hSLE}_{\kappa }(\nu )\) with \(\kappa \in (0,8)\): continuity, reversibility, target-independence, and conditional law characterization. As by-products, we give two applications of these properties. The first one is about the critical Ising interfaces. We prove the convergence of the Ising interface in rectangles with alternating boundary conditions. This result was first proved by Izyurov (Commun Math Phys 337(1):225–252, 2015), and our proof is different. Our method is based on the properties of \(\mathrm {hSLE}\) and is easy to generalize to more complicated boundary conditions and to other models. The second application is the existence of the so-called pure partition functions of multiple SLEs. Such existence was proved for \(\kappa \in (0,8){\setminus } {\mathbb {Q}}\) in Kytölä and Peltola (Commun Math Phys 346(1):237–292, 2016), and it was later proved for \(\kappa \in (0,4]\) in Peltola and Wu (Commun. Math. Phys. 366(2):469–536, 2019). We give a new proof of the existence for \(\kappa \in (0,6]\) using the properties of \(\mathrm {hSLE}\). PubDate: 2020-02-19

Abstract: Abstract Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a d-dimensional system of linear equations or linear differential equations with complexity \({{\,\mathrm{poly}\,}}(\log d)\). While several of these algorithms approximate the solution to within \(\epsilon \) with complexity \({{\,\mathrm{poly}\,}}(\log (1/\epsilon ))\), no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity \({{\,\mathrm{poly}\,}}(\log d, \log (1/\epsilon ))\). PubDate: 2020-02-18

Abstract: Abstract In the framework of Quantum Field Theory, we provide a rigorous, operator algebraic notion of entanglement entropy associated with a pair of open double cones \(O \subset {\widetilde{O}}\) of the spacetime, where the closure of O is contained in \({\widetilde{O}}\). Given a QFT net \({\mathcal {A}}\) of local von Neumann algebras \({\mathcal {A}}(O)\), we consider the von Neumann entropy \(S_{\mathcal {A}}(O, {\widetilde{O}})\) of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras \({\mathcal {A}}(O)\subset {\mathcal {A}}({\widetilde{O}})\) (split property). We show that this canonical entanglement entropy \(S_{\mathcal {A}}(O, {\widetilde{O}})\) is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). To this end, we first study the notion of von Neumann entropy of a closed real linear subspace of a complex Hilbert space, that we then estimate for the local free fermion subspaces. We further consider the lower entanglement entropy \({\underline{S}}_{\mathcal {A}}(O, {\widetilde{O}})\), the infimum of the vacuum von Neumann entropy of \({\mathcal {F}}\), where \({\mathcal {F}}\) here runs over all the intermediate, discrete type I von Neumann algebras. We prove that \({\underline{S}}_{\mathcal {A}}(O, {\widetilde{O}})\) is finite for the local chiral conformal net generated by finitely many commuting U(1)-currents. PubDate: 2020-02-17

Abstract: Abstract We present a construction of cellular BF theory (in both abelian and non-abelian variants) on cobordisms equipped with cellular decompositions. Partition functions of this theory are invariant under subdivisions, satisfy a version of the quantum master equation, and satisfy Atiyah–Segal-type gluing formula with respect to composition of cobordisms. PubDate: 2020-02-17

Abstract: Abstract We show that typical [in the sense of Bonatti and Viana (Ergod Theory Dyn Syst 24(5):1295–1330, 2004) and Avila and Viana (Port Math 64:311–376, 2007)] Hölder and fiber-bunched \(\text {GL}_d(\mathbb {R})\)-valued cocycles over subshifts of finite type are uniformly quasi-multiplicative with respect to all singular value potentials. We prove the continuity of the singular value pressure and its corresponding (necessarily unique) equilibrium state for such cocycles, and apply this result to repellers. Moreover, we show that the pointwise Lyapunov spectrum is closed and convex, and establish partial multifractal analysis on the level sets of pointwise Lyapunov exponents for such cocycles. PubDate: 2020-02-14

Abstract: Abstract We consider the evolution of a quantum particle hopping on a cubic lattice in any dimension and subject to a potential consisting of a periodic part and a random part that fluctuates stochastically in time. If the random potential evolves according to a stationary Markov process, we obtain diffusive scaling for moments of the position displacement, with a diffusion constant that grows as the inverse square of the disorder strength at weak coupling. More generally, we show that a central limit theorem holds such that the square amplitude of the wave packet converges, after diffusive rescaling, to a solution of a heat equation. PubDate: 2020-02-14

Abstract: Abstract Janus and Epimetheus are two moons of Saturn with very peculiar motions. As they orbit around Saturn on quasi-coplanar and quasi-circular trajectories whose radii are only 50 km apart (less than their respective diameters), every four (terrestrial) years the bodies approach each other and their mutual gravitational influence lead to a swapping of the orbits: the outer moon becomes the inner one and vice-versa. This behavior generates horseshoe-shaped trajectories depicted in an appropriate rotating frame. In spite of analytical theories and numerical investigations developed to describe their long-term dynamics, so far very few rigorous long-time stability results on the “horseshoe motion” have been obtained even in the restricted three-body problem. Adapting the idea of Arnol’d (Russ Math Surv 18:85–191, 1963) to a resonant case (the co-orbital motion is associated with trajectories in 1:1 mean motion resonance), we provide a rigorous proof of existence of 2-dimensional elliptic invariant tori on which the trajectories are similar to those followed by Janus and Epimetheus. For this purpose, we apply KAM theory to the planar three-body problem. PubDate: 2020-02-13

Abstract: Abstract We introduce the concept of a holomorphic field theory on any complex manifold in the language of the Batalin–Vilkovisky formalism. When the complex dimension is one, this setting agrees with that of chiral conformal field theory. Our main result concerns the behavior of holomorphic theories under renormalization group flow. Namely, we show that holomorphic theories are one-loop finite. We use this to completely characterize holomorphic anomalies in any dimension. Throughout, we compare our approach to holomorphic field theories to more familiar approaches including that of supersymmetric field theories. PubDate: 2020-02-13

Abstract: Abstract We consider a system of \(N\gg 1\) interacting fermionic particles in three dimensions, confined in a periodic box of volume 1, in the mean-field scaling. We assume that the interaction potential is bounded and small enough. We prove upper and lower bounds for the correlation energy, which are optimal in their N-dependence. Moreover, we compute the correlation energy at leading order in the interaction potential, recovering the prediction of second order perturbation theory. The proof is based on the combination of methods recently introduced for the study of fermionic many-body quantum dynamics together with a rigorous version of second-order perturbation theory, developed in the context of non-relativistic QED. PubDate: 2020-02-11

Abstract: Abstract We establish that, in an appropriate limit, qubits of communication should be regarded as composite resources, decomposing cleanly into independent correlation and transmission components. Because qubits of communication can establish ebits of entanglement, qubits are more powerful resources than ebits. We identify a new communications resource, the zero-bit, which is precisely half the gap between them; replacing classical bits by zero-bits makes teleportation asymptotically reversible. This decomposition of a qubit into an ebit and two zero-bits has wide-ranging consequences including applications to state merging, the quantum channel capacity, entanglement distillation, quantum identification and remote state preparation. The source of these results is the theory of approximate quantum error correction. The action of a quantum channel is reversible if and only if no information is leaked to the environment, a characterization that is useful even in approximate form. However, different notions of approximation lead to qualitatively different forms of quantum error correction in the limit of large dimension. We study the effect of a constraint on the dimension of the reference system when considering information leakage. While the resulting condition fails to ensure that the entire input can be corrected, it does ensure that all subspaces of dimension matching that of the reference are correctable. The size of the reference can be characterized by a parameter \(\alpha \); we call the associated resource an \(\alpha \)-bit. Changing \(\alpha \) interpolates between standard quantum error correction and quantum identification, a form of equality testing for quantum states. We develop the theory of \(\alpha \)-bits, including the applications above, and determine the \(\alpha \)-bit capacity of general quantum channels, finding single-letter formulas for the entanglement-assisted and amortised variants. PubDate: 2020-02-10

Abstract: Abstract Green functions in a quantum field theory can be expanded as bivariate series in the coupling and a scale parameter. The leading logs are given by the main diagonal of this expansion, i.e. the subseries where the coupling and the scale parameter appear to the same power; then the next-to leading logs are listed by the next diagonal of the expansion, where the power of the coupling is incremented by one, and so on. We give a general method for deriving explicit formulas and asymptotic estimates for any \(\hbox {next-to}{}^k\) leading-log expansion for a large class of single scale Green functions. These Green functions are solutions to Dyson–Schwinger equations that are known by previous work to be expressible in terms of chord diagrams. We look in detail at the Green function for the fermion propagator in massless Yukawa theory as one example, and the Green function of the photon propagator in quantum electrodynamics as a second example, as well as giving general theorems. Our methods are combinatorial, but the consequences are physical, giving information on which terms dominate and on the dichotomy between gauge theories and other quantum field theories. PubDate: 2020-02-10

Abstract: Abstract Using the description of multiline queues as functions on words, we introduce the notion of a spectral weight of a word by defining a new weighting on multiline queues. We show that the spectral weight of a word is invariant under a natural action of the symmetric group, giving a proof of the commutativity conjecture of Arita, Ayyer, Mallick, and Prolhac. We give a determinant formula for the spectral weight of a word, which gives a proof of a conjecture of the first author and Linusson. PubDate: 2020-02-10

Abstract: Abstract In this paper, we investigate a family of models for a qubit interacting with a bosonic field. More precisely, we find asymptotic limits of the Hamiltonian as the interaction strength tends to infinity. The main result has two applications. First of all, we show that any self-energy renormalisation scheme similar to that of the Nelson model does not converge for the three-dimensional Spin-Boson model. Secondly, we show that excited states exist in the massive Spin-Boson model for sufficiently large interaction strengths. We are also able to compute the asymptotic limit of many physical quantities. PubDate: 2020-02-05

Abstract: Abstract This article is concerned with properties of delocalization for eigenfunctions of Schrödinger operators on large finite graphs. More specifically, we show that the eigenfunctions have a large support and we assess their \(\ell ^p\)-norms. Our estimates hold for any fixed, possibly irregular graph, in prescribed energy regions, and also for certain sequences of graphs such as N-lifts. PubDate: 2020-02-01

Abstract: Abstract We develop a renormalisation group approach to deriving the asymptotics of the spectral gap of the generator of Glauber type dynamics of spin systems with strong correlations (at and near a critical point). In our approach, we derive a spectral gap inequality for the measure recursively in terms of spectral gap inequalities for a sequence of renormalised measures. We apply our method to hierarchical versions of the 4-dimensional n-component \( \varphi ^4\) model at the critical point and its approach from the high temperature side, and of the 2-dimensional Sine-Gordon and the Discrete Gaussian models in the rough phase (Kosterlitz–Thouless phase). For these models, we show that the spectral gap decays polynomially like the spectral gap of the dynamics of a free field (with a logarithmic correction for the \( \varphi ^4\) model), the scaling limit of these models in equilibrium. PubDate: 2020-02-01