Limit Of Infinite Dimensions Research Articles

Sampling from Archimedean copulas

We develop sampling algorithms for multivariate Archimedean copulas. For exchangeable copulas, where there is only one generating function, we first analyse the distribution of the copula itself, deriving a number of integral representations and a generating function representation. One of the integral representations is related, by a form of convolution, to the distribution whose Laplace transform yields the copula generating function. In the infinite-dimensional limit there is a direct connection between the distribution of the copula value and the inverse Laplace transform. Armed with these results, we present three sampling algorithms, all of which entail drawing from a one-dimensional distribution and then scaling the result to create random deviates distributed according to the copula. We implement and compare the various methods. For more general cases, in which an N-dimensional Archimedean copula is given by N−1 nested generating functions, we present algorithms in which each new variate is drawn conditional only on the value of the copula of the previously drawn variates. We also discuss the use of composite nested and exchangeable copulas for modelling random variates with a natural hierarchical structure, such as ratings and sectors for obligors in credit baskets.

Cellular automata on high-dimensional hypercubes.

The emergence of nontrivial collective behavior is studied in large families of cellular automata rules implemented on high-dimensional hypercubes. Evidence is found that the region of rule space where such macroscopic dynamics exists is well-defined in the infinite-dimension limit.

An infinite dimensional central limit theorem for correlated martingales

An infinite dimensional central limit theorem for correlated martingales

An infinite dimensional central limit theorem for correlated martingales

The paper derives a functional central limit theorem for the empirical distributions of a system of strongly correlated continuous martingales at the level of the full trajectory space. We provide a general class of functionals for which the weak convergence to a centered Gaussian random field takes place. An explicit formula for the covariance is established and a characterization of the limit is given in terms of an inductive system of SPDEs. We also show a density theorem for a Sobolev-type class of functionals on the space of continuous functions.

Antiferromagnetism of almost localized fermions: Evolution from Slater-type to Mott-Hubbard gap

We supplement (and critically overview) the existing extensive analysis of antiferromagnetic solution for the Hubbard model with a detailed discussion of two specific features, namely (i) the evolution of the magnetic (Slater) gap (here renormalized by the electronic correlations) into the Mott-Hubbard or atomic gap, and (ii) a rather weak renormalization of the effective mass by the correlations in the half-filled-band case, which contrasts with that for the paramagnetic case. The mass remains strongly enhanced in the non-half-filled-band case. We also stress the difference between magnetic and non-magnetic contributions to the gap. These results are discussed within the slave boson approach in the saddle-point approximation, in which there appears a non-linear staggered molecular field due to the electronic correlations that leads to the appearance of the magnetic gap. They reproduce correctly the ground-state energy in the limit of strong correlations. A brief comparison with the solution in the limit of infinite dimensions and the corresponding situation in the doubly-degenerate-band case with one electron per atom is also made.

Thermal transport for many-body tight-binding models

We clarify some aspects of the calculation of the thermal transport coefficients. For a tight-binding Hamiltonian we discuss the approximate nature of the charge current and the thermal current obtained by Peierls substitution which is also identical to the equation of motion technique. We address the issue of choosing an appropriate basis for making the Peierls construction for transport calculations. We propose a criteria for finding an optimum Wannier basis where the difference between the exact current and the approximate one is minimum. Using the equations of motion we derive the thermal current for a generalized Hubbard model with density interaction. We identify a part which is the contribution from the long range interactions to the heat current. For the Hubbard model we derive expressions for the transport coefficients in the limit of infinite dimensions.

DIRECT LIMITS OF LOW-RANK REPRESENTATIONS OF CLASSICAL GROUPS

ABSTRACT Using Segal-Bargmann model of the oscillator representation, we demonstrate how to organize unipotent representations of the classical groups , and into compatible families. This construction leads to the irreducible representations of infinite-dimensional direct limit groups , and . We then exhibit some examples of the arising representations and address the question of their admissibility.

Structural glass on a lattice in the limit of infinite dimensions

We construct a mean-field theory for the lattice model of a structural glass and solve it using the replica method and one-step replica symmetry breaking ansatz; this theory becomes exact in the limit of infinite dimensions. Analyzing the stability of this solution we conclude that the metastable states remain uncorrelated in a finite temperature range below the transition, but become correlated at a certain low temperature; below this temperature metastable states bifurcate and the one step replica symmetry breaking solution fails. We find dynamic and thermodynamic transition temperatures as functions of the density, and construct a full thermodynamic description of a typical physical process in which the system is trapped in one metastable state when cooled below the vitrification temperature. We find that for such physical process the entropy and pressure at the glass transition are continuous across the transition, while their temperature derivatives have jumps.

CONSTRUCTION OF A DISPERSION RELATION FROM AN ARBITRARY DENSITY OF STATES

The dispersion relations of energy bands in solids are characterized by their density of states, but a given density of states may originate from various band structures. We show how a spherically symmetric dispersion can be constructed for any one-band density of states. This method is applied to one-, two- and three-dimensional systems. It also serves to establish that any one-band spectrum with finite bandwidth can be obtained from a properly scaled dispersion relation in the limit of infinite dimensions.

Quantum encodings in spin systems and harmonic oscillators

We show that higher-dimensional versions of qubits, or qudits, can be encoded into spin systems and into harmonic oscillators, yielding important advantages for quantum computation. Whereas qubit-based quantum computation is adequate for analyses of quantum vs classical computation, in practice qubits are often realized in higher-dimensional systems by truncating all but two levels, thereby reducing the size of the precious Hilbert space. We develop natural qudit gates for universal quantum computation, and exploit the entire accessible Hilbert space. Mathematically, we give representations of the generalized Pauli group for qudits in coupled spin systems and harmonic oscillators, and include analyses of the qubit and the infinite-dimensional limits.

Analysis of the dynamical cluster approximation for the Hubbard model

We examine a central approximation of the recently introduced dynamical cluster approximation (DCA) by example of the Hubbard model. By both analytical and numerical means we study noncompact and compact contributions to the thermodynamic potential. We show that approximating noncompact diagrams by their cluster analogs results in a larger systematic error as compared to the compact diagrams. Consequently, only the compact contributions should be taken from the cluster, whereas noncompact graphs should be inferred from the appropriate Dyson equation. The distinction between noncompact and compact diagrams persists even in the limit of infinite dimensions. Nonlocal corrections beyond the DCA exist for the noncompact diagrams, whereas they vanish for compact diagrams.

Local performance of the (1 + 1)-ES in a noisy environment

While noise is a phenomenon present in many real world optimization problems, the understanding of its potential effects on the performance of evolutionary algorithms is still incomplete. This paper investigates the effects of fitness proportionate Gaussian noise for a (1 + 1)-ES with isotropic normal mutations on the quadratic sphere in the limit of infinite search-space dimensionality. It is demonstrated experimentally that the results provide a good approximation for finite space dimensionality. It is shown that overvaluation as a result of failure to re-evaluate parental fitness leads to both reduced success probabilities and improved performance. Implications for mutation strength adaptation rules are discussed and optimal re-sampling rates are computed.

Performance analysis of evolution strategies with multi-recombination in high-dimensional [formula omitted]-search spaces disturbed by noise

The presence of noise in real-world optimization problems poses difficulties to optimization strategies. It is frequently observed that evolutionary algorithms are quite capable of succeeding in noisy environments. Intuitively, the use of a population of candidate solutions alongside with some implicit or explicit form of averaging inherent in the algorithms is considered responsible. However, so as to arrive at a deeper understanding of the reasons for the capabilities of evolutionary algorithms, mathematical analyses of their performance in select environments are necessary. Such analyses can reveal how the performance of the algorithms scales with parameters of the problem—such as the dimensionality of the search space or the noise strength—or of the algorithms—such as population size or mutation strength. Recommendations regarding the optimal sizing of such parameters can then be derived. The present paper derives an asymptotically exact approximation to the progress rate of the ( μ/ μ I , λ)-evolution strategy (ES) on a finite-dimensional noisy sphere. It is shown that, in contrast to results obtained in the limit of infinite search space dimensionality, there is a finite optimal population size above which the efficiency of the strategy declines, and that therefore it is not possible to attain the efficiency that can be achieved in the absence of noise by increasing the population size. It is also shown that nonetheless, the benefits of genetic repair and an increased mutation strength make it possible for the multi-parent ( μ/ μ I , λ)-ES to far outperform simple one-parent strategies.

Extended Pegg-Barnett Phase Operator

Vorontsov and Rembovsky pointed out recently a problem that unphysical states appear after an approximate quantum measurement of the Pegg-Barnett phase [Phys. Lett. A 254 (1999), 7; 262 (1999), 486]. To give a solution to this problem, we introduce a quantum phase in the extended number-state space and show that there is no Vorontsov-Rembovsky problem in the extended space. In the infinite-dimensional limit, the present extended formalism gives the same results in almost all cases as the Pegg-Barnett formalism. Also, it is shown that unphysical states can be suppressed by using the projection operator to the physical number space.

Toward a systematic1/dexpansion: Two-particle properties

We present a procedure to calculate 1/d corrections to the two-particle properties around the infinite dimensional dynamical mean field limit. Our method is based on a modified version of the scheme of Ref. onlinecite{SchillerIngersent}}. To test our method we study the Hubbard model at half filling within the fluctuation exchange approximation (FLEX), a selfconsistent generalization of iterative perturbation theory. Apart from the inherent unstabilities of FLEX, our method is stable and results in causal solutions. We find that 1/d corrections to the local approximation are relatively small in the Hubbard model.

Insulator-to-metal crossover induced by local spinfluctuations in strongly correlated systems

We study the simplified Hubbard (SH) model in the presence of atransverse field in the infinite-dimension limit. The relevantone-particle Green's functions of the model are obtained by means of aperturbative treatment of the hopping and of the transverse fieldaround the atomic limit. We consider an analytical solution for theimpurity problem. It is shown that this solution is very accuratein describing the spectral properties of the heavy particles of theSH model for intermediate and strong values of the on-site Coulombinteraction U. We find that for large values of U aninsulator-metal transition takes place as a function of thetransverse field. We analyse the metallic phase through thebehaviour of the density of states and those of the optical conductivityand static resistivity. Our results for the latter quantity agreewith what is observed in experiments on Bi2Sr2CuOy.

Barnett-Pegg formalism of angle operators, revivals, and flux lines

We use the Barnett-Pegg formalism of angle operators to study a rotating particle with and without a flux line. Requiring a finite dimensional version of the Wigner function to be well defined we find a natural time quantization that leads to classical maps from which the arithmetical basis of quantum revivals is seen. The flux line, that fundamentally alters the quantum statistics, forces this time quantum to be increased by a factor of a winding number and determines the homotopy class of the path. The value of the flux is restricted to the rational numbers, a feature that persists in the infinite dimensional limit.

How the New Scale Relativity Theory resolves some quantum paradoxes

It is explicitly shown that within the framework of the New Relativity Theory, some Quantum Mechanical Paradoxes like the Einstein–Rosen Podolsky and the Black Hole Information Loss, are easily resolved. Such New Relativity Theory requires the introduction of an infinite-dimensional quantum space-time as has been shown recently by one of us. This can be viewed as just another way of looking at Feynman's path integral formulation of Quantum Mechanics. Instead of having an infinite-dimensional functional integral over all paths, smooth, forwards and backwards in time, random and fractal, in a finite-dimensional space-time, one has a finite number of paths in an infinite-dimensional quantum space-time. We present a few-lines proof why there is no such thing as an EPR Paradox in this New Relativity theory. The reason is not due to a superluminal information speed but to a divergent information charge density. In the infinite-dimensional limit, due to the properties of gamma functions, the hypervolume enclosed by a D-dim hypersphere, of finite non-zero radius, shrinks to zero: to a hyperpoint, the infinite-dimensional analog to a point. For this reason, Information flows through the infinite-dimensional hypersurface of non-zero radius, but zero size, the hyperpoint, in an instant. In this fashion we imbue an abstract mathematical “point” with a true physical meaning: it is an entity in infinite dimensions that has zero hypervolume at non-zero radius. A plausible resolution of the Information Loss Paradox in Black Holes is proposed.

Transport properties in the boson–fermion model for high- Tc superconductors

We investigate the anomalous temperature and doping dependence of the Hall coefficient in high- T c cuprate superconductors. The study is performed within the framework of the boson–fermion model, which describes a mixture of quasi-free electrons and tightly bound electron pairs, hybridized with each other via a charge–exchange mechanism. The model is analyzed in the limit of infinite dimensions using the non-crossing approximation within the dynamical mean-field theory.

Segregation and charge-density-wave order in the spinless Falicov-Kimball model

The spinless Falicov-Kimball model is solved exactly in the limit of infinite dimensions on both the hypercubic and Bethe lattices. The competition between segregation, which is present for large $U,$ and charge-density-wave order, which is prevalent at moderate $U,$ is examined in detail. We find a rich phase diagram that displays both of these phases. The model also shows nonanalytic behavior in the charge-density-wave transition temperature when $U$ is large enough to generate a correlation-induced gap in the single-particle density of states.