Saddle-point Evaluation Research Articles

Research on the efficient EM modeling method from multilayered anisotropic medium-metal composite targets

The paper presents a high-frequency modeling approach tailored for the electromagnetic (EM) scattering characteristics from electrically large radar targets coated with multi-layered anisotropic mediums (MLAMs). The approach begins by deriving the plane-wave spectrum expression for the incident EM field within MLAMs. It then employs spectral domain full-wave analysis method (SDFWAM) to obtain an analytical representation of the scattering field, further leveraging saddle point evaluation (SPE) to derive asymptotic solutions in the spatial domain. By integrating principles of physical optics (PO) and the tangent plane approximation, the far-field scattering characteristics of target enveloped in the specified medium are efficiently delineated. Validations against standard structure and the Misty satellite model reveal the method's pronounced alignment with the method of moments – finite element method (MoM-FEM) hybrid numerical algorithm, underscoring its notable computational efficiency. Furthermore, in conjunction with scattering sources decomposition technique, the approach is applied to optimize the radar cross-section (RCS) of the Su-57 aircraft, achieving precise and intelligent RCS control at minimal material cost. In conclusion, this research offers pivotal technological and theoretical foundations for EM scattering prediction, stealth design, and performance assessment in radar target domains.

Generating functions for giant graviton bound states

We construct generating functions for operators dual to systems of giant gravitons with open strings attached. These operators have a bare dimension of order N so that the usual methods used to solve the planar limit are not applicable. The generating functions are given as integrals over auxiliary variables, which implement symmetrization and antisymmetrization of the indices of the fields from which the operator is composed. Operators of a good scaling dimension (eigenstates of the dilatation operator) are known as Gauss graph operators. Our generating functions give a simple construction of the Gauss graph operators which were previously obtained using a Fourier transform on a double coset. The new description provides a natural starting point for a systematic frac{1}{N} expansion for these operators as well as the action of the dilatation operator on them, in terms of a saddle point evaluation of their integral representation.

Resistance distance distribution in large sparse random graphs

We consider an Erdős–Rényi random graph consisting of N vertices connected by randomly and independently drawing an edge between every pair of them with probability c/N so that at N → ∞ one obtains a graph of finite mean degree c. In this regime, we study the distribution of resistance distances between the vertices of this graph and develop an auxiliary field representation for this quantity in the spirit of statistical field theory. Using this representation, a saddle point evaluation of the resistance distance distribution is possible at N → ∞ in terms of an 1/c expansion. The leading order of this expansion captures the results of numerical simulations very well down to rather small values of c; for example, it recovers the empirical distribution at c = 4 or 6 with an overlap of around 90%. At large values of c, the distribution tends to a Gaussian of mean 2/c and standard deviation . At small values of c, the distribution is skewed toward larger values, as captured by our saddle point analysis, and many fine features appear in addition to the main peak, including subleading peaks that can be traced back to resistance distances between vertices of specific low degrees and the rest of the graph. We develop a more refined saddle point scheme that extracts the corresponding degree-differentiated resistance distance distributions. We then use this approach to recover analytically the most apparent of the subleading peaks that originates from vertices of degree 1. Rather intuitively, this subleading peak turns out to be a copy of the main peak, shifted by one unit of resistance distance and scaled down by the probability for a vertex to have degree 1. We comment on a possible lack of smoothness in the true N → ∞ distribution suggested by the numerics.

The N=4 SU(N) Super-Yang-Mills Index and Dual AdS Black Holes

The main subjects of this dissertation are indices of the N=4 SU(N) Super-Yang-Mills (SYM) theory, namely the topologically twisted index and the superconformal index. These indices have received a lot of attention since they provide microscopic understanding of AdS_5 black strings and black holes respectively through the AdS/CFT correspondence. In this dissertation, we focus on the field theory side and investigate these indices with a goal of improving the current microscopic understanding of AdS_5 black strings and black holes. As a result, we unveil interesting physics of the 4d indices such as modular properties and a relation to the S^3 partition function of effective Chern-Simons theory, and also make suggestions in the gravity side based on the structure of indices through the AdS/CFT correspondence. First, we study the topologically twisted index of the N=4 SU(N) SYM theory on T^2xS^2. We introduce the Bethe Ansatz (BA) formula that gives the twisted index as a sum over solutions to the Bethe Ansatz Equations (BAE) and categorize various solutions into two groups: standard ones that compose the SL(2,Z) family and non-standard ones that denote all the other BAE solutions including continuous families. Focusing on the contribution from standard BAE solutions, we confirm that it behaves as an elliptic genus with certain modular properties and further investigate its asymptotic behaviors in the Cardy-like limit. Lastly, we review how the twisted index counts the microstates associated with the dual AdS_5 black string entropy in the Cardy-like limit, and discuss missing steps that should be taken care of to validate the microstate counting. Next, we study the superconformal index of the N=4 SU(N) SYM theory. We compute the superconformal index first by saddle point evaluation and then by the BA formula. In due process, we establish a direct relation between the 4d superconformal index and the S^3 partition function of Chern-Simons theory. Then we investigate the phase structure of the superconformal index in the large-N after the Cardy-like limit, which contains partially-deconfined phases distinguished from the previously well-known fully-deconfined/confined phases. Finally, we discuss implications of a partially-deconfined phase, based on the counting of microstates associated with the dual AdS_5 black hole entropy by the superconformal index. [

Matrix integrals & finite holography

We explore the conjectured duality between a class of large N matrix integrals, known as multicritical matrix integrals (MMI), and the series (2m − 1, 2) of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an S2 topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an S2 and a T2 topology using BRST cohomology considerations. Matrix integrals support this finiteness.

Distributed Proximal Algorithms for Multiagent Optimization With Coupled Inequality Constraints

This paper aims to address distributed optimization problems over directed and time-varying networks, where the global objective function consists of a sum of locally accessible convex objective functions subject to a feasible set constraint and coupled inequality constraints whose information is only partially accessible to each agent. For this problem, a distributed proximal-based algorithm, called distributed proximal primal-dual (DPPD) algorithm, is proposed based on the celebrated centralized proximal point algorithm. It is shown that the proposed algorithm can lead to the global optimal solution with a general stepsize, which is diminishing and non-summable, but not necessarily square-summable, and the saddle-point running evaluation error vanishes proportionally to $O(1/\sqrt{k})$, where $k>0$ is the iteration number. Finally, a simulation example is presented to corroborate the effectiveness of the proposed algorithm.

Constraining the effective action by a method of external sources

We propose a novel method of evaluating the effective action, wherein the physical one- and two-point functions are obtained in the limit of non-vanishing external sources. We illustrate the self-consistency of this method by recovering the usual 2PI effective action due to Cornwall, Jackiw and Tomboulis, differing only by the fact that the saddle-point evaluation of the path integral is performed along the extremal quantum, rather than classical, path. As such, this approach is of particular relevance to situations where the dominant quantum and classical paths are non-perturbatively far away from one-another. A pertinent example is the decay of false vacua in radiatively-generated potentials, as may occur for the electroweak vacuum of the Standard Model. In addition, we describe how the external sources may instead be chosen so as to yield the two-particle-point-irreducible (2PPI) effective action of Coppens and Verschelde. Finally, in the spirit of the symmetry-improved effective action of Pilaftsis and Teresi, we give an example of how the external sources can be used to preserve global symmetries in truncations of the 2PI effective action. Specifically, in the context of an O(2) model with spontaneous symmetry breaking, we show that this approach allows the Hartree–Fock approximation to be re-organized, such that the Goldstone boson remains massless algebraically in the symmetry-broken phase and we obtain the correct second-order thermal phase transition.

Solution of the Dynamics of Liquids in the Large-Dimensional Limit.

We obtain analytic expressions for the time correlation functions of a liquid of spherical particles, exact in the limit of high dimensions d. The derivation is long but straightforward: a dynamic virial expansion for which only the first two terms survive, followed by a change to generalized spherical coordinates in the dynamic variables leading to saddle-point evaluation of integrals for large d. The problem is, thus, mapped onto a one-dimensional diffusion in a perturbed harmonic potential with colored noise. At high density, an ergodicity-breaking glass transition is found. In this regime, our results agree with thermodynamics, consistently with the general random first order transition scenario. The glass transition density is higher than the best known lower bound for hard sphere packings in large d. Because our calculation is, if not rigorous, elementary, an improvement in the bound for sphere packings in large dimensions is at hand.

Evaluation of Scattering From Electrically Large and Complex PEC Target Coated With Uniaxial Electric Anisotropic Medium Layer Based on Asymptotic Solution in Spectral Domain

This paper presents a novel and original approach for the evaluation of electromagnetic (EM) scattering from electrically large and complex target coated with uniaxial electric anisotropic medium (UEAM) layer. The evaluation includes two indispensable aspects, the geometry modeling of the target and the determination of the surface currents. Once the equivalent surface currents (ESCs) induced on the outer surface of the coated target are determined, the scattered field could be calculated by the corresponding radiation integrals. First, the outer surface of the arbitrarily shaped target is discretized using flat triangle facets. Second, the ESCs on any facet are deduced from the surface fields, which involve a complicated coupling of type I and type II waves existing in the UEAM. Thus, the determination of the surface fields and the ESCs on the surface of the target with large radius of curvature is the crux of the whole evaluation. By making the tangent plane approximation, used in the physical optics (PO) method, any discretized flat facet could be approximated as an infinite PEC plate coated with a homogenous UEAM layer. Thereafter, the scattering from the infinite coated plate is asymptotically calculated by using the proposed spectral domain method combined with saddle point evaluation (SDM-SPE). Ultimately, comparisons of the proposed asymptotic solution with the reference solutions calculated numerically confirm the validity and efficiency of the proposed SDM-SPE. Numerical results of several canonical and complex targets coated with UEAM layer are given by the proposed approach for the first time.

Interference structure of above-threshold ionization versus above-threshold detachment

Laser-induced electron detachment or ionization of atoms and negative ions is considered. In the context of the saddle-point evaluation of the strong-field approximation (SFA), the velocity maps of the direct electrons (those that do not undergo rescattering) exhibit a characteristic structure due to the constructive and destructive interference of electrons liberated from their parent atoms/ions within certain windows of time. This structure is defined by the above-threshold ionization rings at fixed electron energy and by two sets of curves in momentum space on which destructive interference occurs. The spectra obtained with the SFA are compared with those obtained by numerical solution of the time-dependent Schrödinger equation. For detachment, the agreement is excellent. For ionization, the effect of the Coulomb field is most pronounced for electrons emitted in a direction close to laser polarization, while for near-perpendicular emission the qualitative appearance of the spectrum is unaffected.

Application of Edwards’ statistical mechanics to high-dimensional jammed sphere packings

The isostatic jamming limit of frictionless spherical particles from Edwards' statistical mechanics [Song et al., Nature (London) 453, 629 (2008)] is generalized to arbitrary dimension d using a liquid-state description. The asymptotic high-dimensional behavior of the self-consistent relation is obtained by saddle-point evaluation and checked numerically. The resulting random close packing density scaling ϕ∼d2(-d) is consistent with that of other approaches, such as replica theory and density-functional theory. The validity of various structural approximations is assessed by comparing with three- to six-dimensional isostatic packings obtained from simulations. These numerical results support a growing accuracy of the theoretical approach with dimension. The approach could thus serve as a starting point to obtain a geometrical understanding of the higher-order correlations present in jammed packings.

Black holes and phase-space noncommutativity

We use the solutions of the noncommutative Wheeler-De Witt equation arising from a Kantowski-Sachs cosmological model to compute thermodynamic properties of the Schwarzschild black hole. We show that the noncommutativity in the momentum sector introduces a quadratic term in the potential function of the black hole minisuperspace model. This potential has a local minimum and thus the partition function can be computed by resorting to a saddle point evaluation in the neighbourhood of the minimum. The temperature and the entropy of the black hole are derived, and they have an explicit dependence on the inverse of the momentum noncommutative parameter, $\eta$. Moreover, we study the $t=r=0$ singularity in the noncommutative regime and show that in this case the wave function of the system vanishes in the neighbourhood of $t=r=0$.

Exact results in a slave boson saddle point approach for a strongly correlated electron model

We revisit the Kotliar-Ruckenstein (KR) slave boson saddle point evaluation for a two-site correlated electron model. As the model can be solved analytically, it is possible to compare the KR saddle point results with the exact many-particle levels. The considered two-site cluster mimics an infinite-$U$ single-impurity Anderson model with a nearest-neighbor Coulomb interaction: one site is strongly correlated with an infinite local Coulomb repulsion, which hybridizes with the second site, on which the local Coulomb repulsion vanishes. Making use of the flexibility of the representation, we introduce appropriate weight factors in the KR saddle point scheme. Ground-state and all excitation levels agree with the exact diagonalization results. Thermodynamics and correlation functions may be recovered in a suitably renormalized saddle point evaluation.

Double-spin asymmetries for small-[formula omitted] Drell–Yan pair production in transversely polarized [formula omitted] collisions

We discuss the Drell–Yan process at a measured transverse-momentum QT of the produced lepton pair in collisions of transversely polarized protons and antiprotons, to be observed at the proposed spin experiments at GSI. The large logarithmic contributions from multiple soft gluon emission, accompanying the Drell–Yan mechanism at small QT, are resummed to all orders in QCD perturbation theory up to next-to-leading logarithmic (NLL) accuracy. Numerical evaluation shows the impact of the NLL as well as LL effect on the dilepton QT spectra. For the corresponding QT-dependent spin asymmetry ATT(QT), the LL effect gives significant modification while the NLL effect is marginal, leading to QCD prediction that ATT(QT) at GSI is flat at small and moderate QT and almost equals the conventional asymmetry ATT associated with the QT-integrated cross sections. This flat behavior in turn allows us to use analytic saddle-point evaluation of the resummation formula in the limit QT→0, not only to obtain quantitative estimate of ATT(QT), but also to clarify mechanisms behind the relation ATT(QT)≃ATT characteristic of pp¯ collisions at GSI.

Does multifractal theory of turbulence have logarithms in the scaling relations?

The multifractal theory of turbulence uses a saddle-point evaluation in determining the power-law behaviour of structure functions. Without suitable precautions, this could lead to the presence of logarithmic corrections, thereby violating known exact relations such as the four-fifths law. Using the theory of large deviations applied to the random multiplicative model of turbulence and calculating subdominant terms, we explain here why such corrections cannot be present.

Phase-space path-integral calculation of the Wigner function

The Wigner function W(q,p) is formulated as a phase-space path integral, whereby its sign oscillations can be seen to follow from interference between the geometrical phases of the paths. The approach has similarities to the path-centroid method in the configuration-space path integral. Paths can be classified by the mid-point of their ends; short paths where the mid-point is close to (q,p) and which lie in regions of low energy (low P function of the Hamiltonian) will dominate, and the enclosed area will determine the sign of the Wigner function. As a demonstration, the method is applied to a sequence of density matrices interpolating between a Poissonian number distribution and a number state, each member of which can be represented exactly by a discretized path integral with a finite number of vertices. Saddle point evaluation of these integrals recovers (up to a constant factor) the WKB approximation to the Wigner function of a number state.

Plane-wave scattering-matrix formulation for two interfaces plus a scatterer (uniplanar case)

Kern's plane-wave scattering-matrix formulation is extended to treat the case of two interfaces plus a scatterer imbedded in the second region. This formulation can treat different shapes for the scatterer and only requires two different evaluations depending upon the location of the observer: 1) in the near field, a two-dimensional (2-D) fast Fourier transform (FFT) (one-dimensional (1-D) FFT is treated by an analytical integration), and 2) in the far-field an asymptotic evaluation of the integral. These two regimes are in contrast to the Sommerfeld integral approximations where different approximations are required for various parameter ranges (quasistatic, saddle-point evaluation for the radiation field, saddle point evaluation for the surface field when the saddle point is near a pole, and the lateral wave field evaluated using a uniform asymptotic evaluation for the integrals).

Density-functional theory and average-atom model formulated in terms of functional integrals

Functional integrals and the notion of Legendre transformation are used to formulate the average-atom model and the density-functional theory in the same formalism, stressing their common theoretical basis. Particularly, the Kohn and Sham equations are proven to appear naturally by performing a saddle-point evaluation of a specific functional integral. Exact expressions for the exchange-correlation terms in the grand potential at finite temperature are derived.

Loop quantum mechanics and the fractal structure of quantum spacetime

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of “loop quantum mechanics”, we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate “evaporates”, and what is left behind is a “vacuum” characterized by an effective fractal geometry.

Range-dependent acoustic radiation from line-driven plates above the bending coincidence frequency

The leaky waves generated on plates driven in bending above the bending coincidence frequency radiate as extended sources; therefore, the standard saddle point evaluation of the radiated pressure, which yields a far-field result, may not be appropriate at the range of interest. The calculation can be modified by removing the contribution of the leaky waves from the original integrand, evaluating the remaining regular part by the saddle point method, and evaluating the effect of the leaky waves in terms of error functions and residue. Results showing range dependence are presented for radiation by a steel plate in water driven by normal and tangential line forces. The correctness of the results is made plausible by comparing independent calculations of acoustic power passing through a semicircle at finite range.