Asymptotic Behavior of Inflated Lattice Polygons

We study the inflated phase of two dimensional lattice polygons with fixed perimeter N and variable area, associating a weight exp [pA−Jb] to a polygon with area A and b bends. For convex and column-convex polygons, we calculate the average area for positive values of the pressure. For large pressures, the area has the asymptotic behaviour , where , and ρ<1. The constant K(J) is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for J≠0. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.     

On the area of square lattice polygons

We consider the generating function of self-avoiding square lattice polygons grouped by both area and perimeter. The generating function for polygons of arean is found to diverge atx _c =0.251834, with an exponent of zero. The mean perimeter of polygons with arean is found to be proportional ton, while the mean area of polygons with perimetern is found to be proportional ton ^1.5.     

Vector potential gauge for superconducting regular polygons

An approach to the Ginzburg-Landau problem of superconducting polygons is developed, based on the exact fulfillment of superconducting boundary conditions along the boundary of the sample. To this end an analytical gauge transformation for the vector potential A is found which gives A _n = 0 for the normal component along the boundary line of an arbitrary regular polygon. The use of the new gauge reduces the Ginzburg-Landau problem of superconducting polygons in external magnetic fields to an eigenvalue problem in a basis set of functions obeying Neumann boundary conditions. The advantages of this approach, especially for low magnetic fields, are illustrated and novel vortex patterns are obtained which can be probed experimentally. Received 28 February 2002 and Received in final form 12 April 2002 Published online 6 June 2002     

The Pentagram Map: A Discrete Integrable System

The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from \mathbb Z{\mathbb Z} into \mathbbRP²{{\mathbb{RP}}^2} that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation. A research announcement of this work appeared in [16].     

The Generation of Random Equilateral Polygons

Freely jointed random equilateral polygons serve as a common model for polymer rings, reflecting their statistical properties under theta conditions. To generate equilateral polygons, researchers employ many procedures that have been proved, or at least are believed, to be random with respect to the natural measure on the space of polygonal knots. As a result, the random selection of equilateral polygons, as well as the statistical robustness of this selection, is of particular interest. In this research, we study the key features of four popular methods: the Polygonal Folding, the Crankshaft Rotation, the Hedgehog, and the Triangle Methods. In particular, we compare the implementation and efficacy of these procedures, especially in regards to the population distribution of polygons in the space of polygonal knots, the distribution of edge vectors, the local curvature, and the local torsion. In addition, we give a rigorous proof that the Crankshaft Rotation Method is ergodic.     

Spatial variations of the mean and statistical quantities in the thermal boundary layers of turbulent convection

We have studied the dynamics of the contact line of a viscous liquid on a solid substrate with macroscopic random defects. We have first characterized the friction force f₀ at microscopic scale for a substrate without defects; f₀ is found to be a strongly nonlinear function of the velocity U of the contact line. In presence of macroscopic defects, we find that the applied force F(U) is simply shifted with respect to f₀(U) by a constant: we do not observe any critical behavior at the depinning transition. The only observable effect of the substrate disorder is to increase the hysteresis. We have also performed realistic numerical simulation of the motion of the contact line. Using the same values of the parameters as in the experiment, we find that the experimental data is qualitatively well reproduced. In light of experimental and numerical results, we discuss the possibility of measuring a true critical behavior.Received: 6 October 2003, Published online: 19 February 2004PACS: 46.65. + g Random phenomena and media - 64.60.Ht Dynamic critical phenomena - 68.08.Bc Wetting     

Subordination pathways to fractional diffusion

The uncoupled Continuous Time Random Walk (CTRW) in one space-dimension and under power law regime is splitted into three distinct random walks: (rw ₁), a random walk along the line of natural time, happening in operational time; (w ₂), a random walk along the line of space, happening in operational time; (rw ₃), the inversion of (rw ₁), namely a random walk along the line of operational time, happening in natural time. Via the general integral equation of CTRW and appropriate rescaling, the transition to the diffusion limit is carried out for each of these three random walks. Combining the limits of (rw ₁) and (rw ₂) we get the method of parametric subordination for generating particle paths, whereas combination of (rw ₂) and (rw ₃) yields the subordination integral for the sojourn probability density in space - time fractional diffusion.     

On functional determinants of laplacians in polygons and simplicial complexes

The functional determinant of an elliptic operator with positive, discrete spectrum may be defined ase ^–Z' (0), whereZ(s), the zeta function, is the sum analytically continued ins. In this paperZ'(0) is calculated for the Laplace operator with Dirichlet boundary conditions inside polygons with the topology of a disc in the Euclidean plane. Our results are complementary to earlier investigations of the determinants on smooth surfaces with smooth boundaries. Our expression can be viewed as the energy for a system of static point particles, corresponding to the corners of the polygon, with self-energy and pair interaction energy. We have completely explicit closed expressions for triangles and regular polygons with an arbitrary number of sides. Among these, there are five special cases (three triangles, the square and the circled), where theZ'(0) are known by other means. One special case fixes an integration constant, and the other provide four independent analytical checks on our calculation.     

Singularity Dominated Strong Fluctuations for Some Random Matrix Averages

The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval (0,1) of the real line respectively. The averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2 diverges, for 2 –1, at points approaching the eigenvalue support. Using the theory of generalized hypergeometric functions based on Jack polynomials, the functional form of the leading asymptotic behaviour is established rigorously. In the circular ensemble case this confirms a conjecture of Berry and Keating.     

Measurement of low retardation using reflective optical system based on the photoelastic principle

A new measuring technique that can measure retardation and can output magnitude and direction of plane stress in each glass of a panel composed of double transparent pieces of glass has been developed using reflective confocal optics. The linear polarized probe beam is incident to the glass and we can detect a reflected beam converted to orthogonal polarization caused by the photoelastic phenomenon. Using the high extinction ratio (10⁶) beam-displacing prism as a polarization discriminator, we can measure the photoelasticity by rotating the polarization of the probe beam from 0 to π rad without disturbing the optical axis. This system has the ability to measure retardation. The lowest one is estimated as nearly 0.066 nm for 700 μm thickness glass which corresponds to 0.03 MPa stress from our calibration line.     

均匀带电平面折线在空间产生的电场及其可视化

推导了均匀带电线段在二维平面中产生的电势的简要公式,进而推导了在三维直角坐标系中的电势公式,形成了带电折线(包括任意多边形)的电势简要公式.根据电势与电场强度之间的关系,列出了场强的表达式.将公式无量纲化,利用MATLAB指令,绘制了各种形状的带电多边形和折线的等势面和三维电场线,并显示了电场在空间的分布规律.     

FT-EPR with a Nonresonant Probe: Use of a Truncated Coaxial Line

A truncated transmission line probe (TLP) has been utilized to excite and detect time domain responses after pulsed excitation in electron paramagnetic resonance (EPR) spectroscopic experiments in the frequency range 200–400 MHz. The TLP device is a modified short-circuited coaxial line, which allows the irradiation of the sample by the traveling waveB₁fields in the frequency range of kilohertz to 30 GHz. In EPR studies at 300 MHz carrier frequency, with 10 W incident power, a 45° pulse is 45 ns in duration. This corresponds to a 0.9-GB₁field. Using the TLP, time-domain responses from the solidN-methyl pyridiniumtetra-cyanoquinodimethane (TCNQ) were collected at 200, 250, 300, and 350 MHz, with the range limited by the amplifiers. In addition two tubes containing TCNQ placed side-by-side vertically along the axis of the probe were used to collect time domain responses in the presence of magnetic field gradients to test the feasibility of two-dimensional imaging using a TLP. The magnetic field gradient was steered in thexzplane and 36 projections were collected at 5° intervals. Using filtered back-projection image reconstruction, the two-dimensional spatial image in thexzplane was obtained at good resolution.     

Transforming Fixed-Length Self-avoiding Walks into Radial SLE<Subscript>8/3</Subscript>

We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE_8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE_8/3. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo simulations of the SAW. Using the conjectured relationship between the SAW and radial SLE_8/3, our simulations give estimates for both the interior and boundary scaling exponents. The values we obtain are within a few hundredths of a percent of the conjectured values.     

Broken supersymmetry in the matrix model on a circle

We consider the discretization of aD=2 surface using polygons. We map the surface onto superspace and integrate over surfaces of arbitrary genus, obtaining a discretized version of the Green-Schwarz string inD=1. Taking an unusual critical limit of the supersymmetric matrix model involved, we construct exact solutions, to all perturbative orders, for the discretized superstring in one dimension, both when the target space is a real line and when the theory is represented in terms of matrix variables on a circle of finite radius. We comment on the behavior of the compactfied perturbative expansion under duality transformations.BITNET: BELLUCCI at IRMLNF     

Random Line Tessellations of the Plane: Statistical Properties of Many-Sided Cells

We consider a family of random line tessellations of the Euclidean plane introduced in a more formal context by Hug and Schneider (Geom. Funct. Anal. 17:156, 2007) and described by a parameter α≥1. For α=1 the zero-cell (that is, the cell containing the origin) coincides with the Crofton cell of a Poisson line tessellation, and for α=2 it coincides with the typical Poisson-Voronoi cell. Let p _n (α) be the probability for the zero-cell to have n sides. We construct the asymptotic expansion of log p _n (α) up to terms that vanish as n→∞. Our methods are nonrigorous but of the kind commonly accepted in theoretical physics as leading to exact results. In the large-n limit the cell is shown to become circular. The circle is centered at the origin when α>1, but gets delocalized for the Crofton cell, α=1, which is a singular point of the parameter range. The large-n expansion of log p _n (1) is therefore different from that of the general case and we show how to carry it out. As a corollary we obtain the analogous expansion for the typical n-sided cell of a Poisson line tessellation.     

The Stochastic Traveling Salesman Problem: Finite Size Scaling and the Cavity Prediction

We study the random link traveling salesman problem, where lengths l _ij between city i and city j are taken to be independent, identically distributed random variables. We discuss a theoretical approach, the cavity method, that has been proposed for finding the optimum tour length over this random ensemble, given the assumption of replica symmetry. Using finite size scaling and a renormalized model, we test the cavity predictions against the results of simulations, and find excellent agreement over a range of distributions. We thus provide numerical evidence that the replica symmetric solution to this problem is the correct one. Finally, we note a surprising result concerning the distribution of k th-nearest neighbor links in optimal tours, and invite a theoretical understanding of this phenomenon.     

Two-dimensional power-type electronic spectrographs with a symmetry plane

The electronic and optical characteristics of 2D electric fields with a complex potential of the type Ω = i(x + iy) ⁿ , where n is a real number, are investigated. Particle dynamics is studied in the symmetry plane and in its neighborhood for constructing an effective spectrograph of electron flows. It is shown that in the range of exponents 0 < n < 1, spatial focusing in the angles of incidence of conical bunches is effected in the system, which has second order in the symmetry plane and at least the first order across it. The line of images of a point source (focal line) is a straight line lying in the symmetry plane, the focusing order being independent of particle energy W. Thus, the spectrographic principle holds, and partial electron fluxes can be detected simultaneously by a position-sensitive detector in a wide range of energy variation. The electrode configuration of these systems is quite simple and can be used in practice for constructing spectrographs. The prospects of application of such spectrographs in energy analysis are considered.     

Large Deviations for the Fermion Point Process Associated with the Exponential Kernel

For the fermion point process on the whole complex plane associated with the exponential kernel , we show the central limit theorem for the random variable ξ(D _r , the number of points inside the ball D _r of radius r, as r → ∞ and we establish the large deviation principle for the random variables {r ⁻²ξ (D _r ), r > 0}.     

The bicritical phase diagram of two-dimensional antiferromagnets with and without random fields

Neutron scattering measurements have been made of the phase diagrams of the nearly two-demensional antiferromagnets Rb₂MnF₄ and Rb₂Mn_0.7Mg_0.3F₄ in a magnetic field applied along thec-axis. In Rb₂MnF₄ there is at low temperatures a spin-flop phase at fields above 5.5 T which has long range order. The observation of true long range order rather than the algebraic decay of the order characteristic of the two-dimensional XY model is presumably due to subtle anisotropy effects in the plane as well as weak three-dimensional coupling. The phase boundaries of the uniaxial and transverse phases are shown to be consistent with renormalization group predictions for two-dimensional systems. The two lines become exponentially close to each other at low temperatures. The weak three-dimensional coupling moves the bicritical point fromT=0 to a non-zero temperature. The situation is more complex in Rb₂Mn_0.7Mg_0.3F₄ because of Ising random field effects. At low fields we observe typical random field metastable behavior with a sharp metastability boundary and a gange of length scales which are time independent below that boundary. At higher fields there are substantial uniaxial fluctuations. The transverse phase boundary and the metastability line appear to intercept atT=0 showing that the random field fluctuations do have a large effect on the phase diagram. The theory of the phase diagrams has been extended to include the random field fluctuations and good agreement is obtained with the observed transverse phase boundary. Unfortunately, there is as yet no theory of the metastable uniaxial phase with which to compare our results.     

SU(1, 1) Random Polynomials

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1, 1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1, 1) random polynomial of degree N are concentrated in a narrow annulus of the order of N ^–1 around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit N. Our results are supported through various numerical simulations. We then extend results of Hannay⁽¹⁾ and Bleher et al. ⁽²⁾ to derive different formulae for correlations between zeros of the SU(1, 1) random analytic functions, by applying the generalized Kac–Rice formula. We express the correlation functions in terms of some Gaussian integrals, which can be evaluated combinatorially as a finite sum over Feynman diagrams or as a supersymmetric integral. Due to the SU(1, 1) symmetry, the correlation functions depend only on the hyperbolic distances between the points on the unit disk, and we obtain an explicit formula for the two point correlation function. It displays quadratic repulsion at small distances and fast decay of correlations at infinity. In an appendix to the paper we evaluate correlations between the outer zeros |z _j |>1 of the SU(1, 1) random polynomial, and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity.