Saddle Point Analysis Research Articles

The Spiral of Life

High-energy photoionization driven by short and circularly-polarized laser pulses is studied in the framework of the relativistic strong-field approximation. The saddle-point analysis of the integrals defining the probability amplitude is used to determine the general properties of the probability distributions. Additionally, an approximate solution to the saddle-point equation is derived. This leads to the concept of the three-dimensional spiral of life in momentum space, around which the ionization probability distribution is maximum. We demonstrate that such spiral is also obtained from a classical treatment.

All point correlation functions in SYK

Large N melonic theories are characterized by two-point function Feynman diagrams built exclusively out of melons. This leads to conformal invariance at strong coupling, four-point function diagrams that are exclusively ladders, and higher-point functions that are built out of four-point functions joined together. We uncover an incredibly useful property of these theories: the six-point function, or equivalently, the three-point function of the primary O(N ) invariant bilinears, regarded as an analytic function of the operator dimensions, fully determines all correlation functions, to leading nontrivial order in 1/N , through simple Feynman-like rules. The result is applicable to any theory, not necessarily melonic, in which higher-point correlators are built out of four-point functions. We explicitly calculate the bilinear three-point function for q-body SYK, at any q. This leads to the bilinear four-point function, as well as all higher-point functions, expressed in terms of higher-point conformal blocks, which we discuss. We find universality of correlators of operators of large dimension, which we simplify through a saddle point analysis. We comment on the implications for the AdS dual of SYK.

Studies on convergence and scaling law of Thomson backscattering spectra in strong fields**Project supported by the National Natural Science Foundation of China (Grant Nos. 11475026 and 11175023).

With the saddle point analysis method for the Bessel function structure and property, the convergence problem and the scaling laws of Thomson backscattering spectra are solved and studied in both cases that are for the plane wave laser field without and with applied external constant magnetic field. Some unclear points appeared in previous work are clarified. The extension of the method to a general situation for the laser field with an arbitrary polarization is discussed. We also make a simple analysis and discussion about the optimal spectra dependence of field parameters and its implication to practical applications.

Cut-off phenomenon in the uniform plane Kac walk

We consider an analogue of the Kac random walk on the special orthogonal group $\mathrm{SO}(N)$, in which at each step a random rotation is performed in a randomly chosen 2-plane of $\mathbb{R}^{N}$. We obtain sharp asymptotics for the rate of convergence in total variance distance, establishing a cut-off phenomenon in the large $N$ limit. In the special case where the angle of rotation is deterministic, this confirms a conjecture of Rosenthal [Ann. Probab. 22 (1994) 398–423]. Under mild conditions, we also establish a cut-off for convergence of the walk to stationarity under the $L^{2}$ norm. Depending on the distribution of the randomly chosen angle of rotation, several surprising features emerge. For instance, it is sometimes the case that the mixing times differ in the total variation and $L^{2}$ norms. Our estimates use an integral representation of the characters of the special orthogonal group together with saddle point analysis.

Above-threshold ionization of helium in the long-wavelength regime: Examining the single-active-electron approximation and the two-electron strong-field approximation

We investigate high-order above-threshold ionization of model helium in the long-wavelength regime up to 2400 nm by solving the two-electron time-dependent Schr\odinger equation in one dimension. To bypass the difficulty of solving the multielectron time-dependent Schr\odinger equation with the long-wavelength laser interaction, we revisit and examine two typically used theoretical methods: the single-active-electron approximation and the strong-field approximation. For the description of the high-energy rescattered electrons in the ground-state ionic channel, the single-active-electron approximation performs better with increasing ponderomotive energy. Single ionization in the excited-state ionic channels, in general, has much weaker spectral intensity than that in the ground-state ionic channel. The above-threshold-ionization cutoffs in the excited-state ionic channels are clear signatures of two-electron dynamics, which cannot be explained within the single-active-electron approximation. By applying the two-electron strong-field approximation including rescattering and a saddle-point method analysis, we explain the channel-resolved cutoffs, and relate them to elastic and inelastic rescattering processes.

High-energy ionization by relativistically intense laser pulses

High-energy ionization by relativistically intense, laser pulses is investigated within the relativistic strong-field approximation. It is shown that the momentum distribution of photoelectrons is focused around a spiral, the analytical form of which is derived from the saddle-point analysis of probability amplitudes. The geometry of this spiral allows one to clearly interpret the exact numerical results of the ionization probability distributions and to identify the photoelectron momentum domains in which the interference- and the interference-free patterns are observed. It is also demonstrated that attosecond single-electron wave packets can be created in this process. Thus, it opens new research opportunities for the four-dimensional electron diffraction experiments, which can be conducted in large-scale laser facilities such as ELI or XCELS.

Study on Operation of a Surface-Wave Oscillator Around the $\pi$ -Point Region

We numerically and experimentally studied the operation of a W-band (75–110 GHz) surface-wave oscillator driven by the electron beam energy of less than 100 keV. Two numerical methods of the real wavenumber analysis and the saddle point analysis are studied to examine the operation of the oscillator. We analyzed the Cherenkov instability for the interaction of the space charge modes and the structure modes EH01. We showed that the instabilities are of absolute instabilities when the oscillator operates around the $\pi $ -point region. In the experiment, radiations due to the absolute instability are detected around the $\pi $ -point region when the length of the slow-wave structure (SWS) is short enough. On the other hand, the absolute instability generates no meaningful radiation in the $\pi $ -point region for the long length of SWS.

ROBUST UTILITY MAXIMIZATION WITH LÉVY PROCESSES

AbstractWe study a robust portfolio optimization problem under model uncertainty for an investor with logarithmic or power utility. The uncertainty is specified by a set of possible Lévy triplets, that is, possible instantaneous drift, volatility, and jump characteristics of the price process. We show that an optimal investment strategy exists and compute it in semi‐closed form. Moreover, we provide a saddle point analysis describing a worst‐case model.

Total variation distance and the Erdős–Turán law for random permutations with polynomially growing cycle weights

Nous nous intéressons à un modèle de permutations aléatoires de $n$ éléments, avec des poids polynomiaux en les longueurs des cycles, qui a été étudié notamment par Ercolani et Ueltschi. En utilisant l’analyse des points selles des transformées de Fourier, nous montrons que la distance en variation totale entre la suite des nombres de cycles de tailles $1,2,\ldots,b$ et une certaine suite $(Z_{1},Z_{2},\ldots,Z_{b})$ de variables de Poisson indépendantes, converge vers $0$ quand $n\to\infty$ si et seulement si $b=o(\ell)$, où $\ell$ désigne la longueur d’un cycle typique de ce modèle. À l’aide de ce résultat, nous établissons ensuite un théorème central limite pour l’ordre de la permutation, étendant ainsi la loi d’Erdős–Turán. Enfin, nous démontrons le caractère brownien pour la limite des petits cycles.

Exact WKB analysis of N $$ \mathcal{N} $$ = 2 gauge theories

We study N = 2 supersymmetric gauge theories with gauge group SU(2) coupled to fundamental flavours, covering all asymptotically free and conformal cases. We re-derive, from the conformal field theory perspective, the differential equations satisfied by omega deformed instanton partition functions. We confirm their validity at leading order in one of the deformation parameters via a saddle-point analysis of the partition function. In the semi-classical limit we show that these differential equations take a form amenable to exact WKB analysis. We compute the monodromy group associated to the differential equations in terms of deformed and Borel resummed Seiberg-Witten data. For each case, we study pairs of Stokes graphs that are related by flips and pops, and show that the monodromy groups allow one to confirm the Stokes automorphisms that arise as the phase of the deformation parameter is varied. Finally, we relate the Borel resummed monodromies with the traditional Seiberg-Witten variables in the semi-classical limit.

Effect of elliptical polarization of driving field on high-order-harmonic generation in semiconductor ZnO

High-order-harmonic generation in ZnO crystal exposed to the elliptically polarized midinfrared light is theoretically investigated upon scanning the ellipticity. We find the normalized yield of each harmonics in the plateau decreases in a form that can be well fitted to Gaussian dependence as increasing the ellipticity of driving pulse. We demonstrate primarily that the sensitivity for the ellipticity dependence of the interband harmonics above the band gap is determined by both the harmonic order and the intensity of driving light, which is interpreted in terms of the semiclassical saddle-point analysis. The effect of more laser pulse parameters on the ellipticity dependence is also investigated. Furthermore, it is shown that the ellipticity dependence of each harmonics is strongly affected by the variation of the energy band in the global and the local way. This shows that high-order-harmonic generation by an elliptically polarized driving field carries the significant signature of the band structure in solids.

Asymptotics of correlation functions of the Heisenberg-Ising chain in the easy-axis regime

We analyse the long-time large-distance asymptotics of the longitudinal correlation functions of the Heisenberg-Ising chain in the easy-axis regime. We show that in this regime the leading asymptotics of the dynamical two-point functions is entirely determined by the two-spinon contribution to their form factor expansion. Its explicit form is obtained from a saddle-point analysis of the corresponding double integral. It describes the propagation of a wave front with velocity which is found to be the maximal possible group velocity. Like in wave propagation in dispersive media the wave front is preceded by a precursor running ahead with velocity . As a special case we obtain the explicit form of the asymptotics of the auto-correlation function.

The Pearcey integral in the highly oscillatory region

We consider the Pearcey integral P(x, y) for large values of |y| and bounded values of |x|. The integrand of the Pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is complicated. Then we consider here the modified saddle point method introduced in [Lopez, Pérez and Pagola, 2009] [4]. With this method, the analysis is simpler and it is possible to derive a complete asymptotic expansion of P(x, y) for large |y|. The asymptotic analysis requires the study of three different regions for argy separately. In the three regions, the expansion is given in terms of inverse powers of y2/3 and the coefficients are elementary functions of x. The accuracy of the approximation is illustrated with some numerical experiments.

Spatial shape of avalanches in the Brownian force model

.We study the Brownian force model (BFM), a solvable model of avalanche statistics for an interface, in a general discrete setting. The BFM describes the overdamped motion of elastically coupled particles driven by a parabolic well in independent Brownian force landscapes. Avalanches are defined as the collective jump of the particles in response to an arbitrary monotonous change in the well position (i.e. in the applied force). We derive an exact formula for the joint probability distribution of these jumps. From it we obtain the joint density of local avalanche sizes for stationary driving in the quasi-static limit near the depinning threshold. A saddle-point analysis predicts the spatial shape of avalanches in the limit of large aspect ratios for the continuum version of the model. We then study fluctuations around this saddle point, and obtain the leading corrections to the mean shape, the fluctuations around the mean shape and the shape asymmetry, for finite aspect ratios. Our results are finally confronted to numerical simulations.

The distribution of Euler–Kronecker constants of quadratic fields

We investigate the distribution of large positive (and negative) values of the Euler–Kronecker constant γQ(D) of the quadratic field Q(D) as D varies over fundamental discriminants |D|≤x. We show that the distribution function of these values is very well approximated by that of an adequate probabilistic random model in a large uniform range. The main tools are an asymptotic formula for the Laplace transform of γQ(D) together with a careful saddle point analysis.

Theory of high-harmonic generation in solids

High-harmonic generation (HHG) in bulk crystals exposed to intense mid-infrared lasers with photon energies below the bandgap is investigated theoretically. A three dimensional, two-band model that considers both interband and intraband currents is used. It is shown that the interband current is the dominant mechanism for HHG in solids. A physical interpretation of interband HHG – similar to atomic HHG – is provided by saddle point analysis. The effects of dephasing time and driving field wavelength on the harmonic specrum are investigated.

Theoretical analysis of high-harmonic generation in solids.

We investigate theoretically high-harmonic generation (HHG) in bulk crystals exposed to intense midinfrared lasers with photon energies smaller than the band gap. The two main mechanisms, interband and intraband HHG, are explored. Our analysis indicates that the interband current neglected so far is the dominant mechanism for HHG. Saddle point analysis in the Keldysh limit yields an intuitive picture of interband HHG in solids similar to atomic HHG. Interband and intraband HHG exhibit a fundamentally different wavelength dependence. This signature can be used to experimentally distinguish between the two mechanisms in order to verify their importance.

Collective excitations in a large-dmodel for graphene

We consider a model of Dirac fermions coupled to flexural phonons to describe a graphene sheet fluctuating in dimension $2+d$. We derive the self-consistent screening equations for the quantum problem, exact in the limit of large $d$. We first treat the membrane alone, and work out the quantum to classical, and harmonic to anharmonic crossover. For the coupled electron-membrane problem we calculate the dressed two-particle propagators of the elastic and electron interactions and find that it exhibits a collective mode which becomes unstable at some wave-vector $q_{\rm c}$ for large enough coupling $g$. The saddle point analysis, exact at large $d$, indicates that this instability corresponds to spontaneous and simultaneous appearance of gaussian curvature and electron puddles. The relevance to ripples in graphene is discussed.

Space-time description of strong-field ionization and high-order-harmonic generation

We develop the spatiotemporal description of matter-field interaction within the strong-field approximation. We show that the space-time form of the ionized wave function has analogies with the diffraction phenomenon, allowing for the definition of two different regimes: Fresnel and Fraunhofer. We demonstrate that the standard saddle-point analysis corresponds to the paraxial approximation of the Fraunhofer case. The Fresnel number therefore appears as a useful parameter to characterize the validity of the saddle-point approach. We give a closed formula for the ionized wave function beyond the standard saddle-point analysis that takes the form of a chirped Volkov wave. We apply our results to the study of high-order-harmonic generation, demonstrating that the saddle-point approximation breaks down for extended systems, i.e., when the Fresnel number approaches or is above the unity. As a simple example, we analyze the harmonic generation of dissociating ${H}_{2}^{+}$ and demonstrate the Fresnel number as a useful parameter to determine the accuracy of the semiclassical saddle-point approach.

Analytic combinatorics in several variables

This book is the first to treat the analytic aspects of combinatorial enumeration from a multivariate perspective. Analytic combinatorics is a branch of enumeration that uses analytic techniques to estimate combinatorial quantities: generating functions are defined and their coefficients are then estimated via complex contour integrals. The multivariate case involves techniques well known in other areas of mathematics but not in combinatorics. Aimed at graduate students and researchers in enumerative combinatorics, the book contains all the necessary background, including a review of the uses of generating functions in combinatorial enumeration as well as chapters devoted to saddle point analysis, Groebner bases, Laurent series and amoebas, and a smattering of differential and algebraic topology. All software along with other ancillary material can be located via the book Web site, http://www.cs.auckland.ac.nz/~mcw/Research/mvGF/asymultseq/ACSVbook/.