Some Partial Results on the Convergence of Loop-Erased Random Walk to SLE(2) in the Natural Parametrization

We outline a strategy for showing convergence of loop-erased random walk on the $\mathbb{Z}^{2}$ square lattice to SLE(2), in the supremum norm topology that takes the time parametrization of the curves into account. The discrete curves are parametrized so that the walker moves at a constant speed determined by the lattice spacing, and the SLE(2) curve has the recently introduced natural time parametrization. Our strategy can be seen as an extension of the one used by Lawler, Schramm, and Werner to prove convergence modulo time parametrization. The crucial extra step is showing that the expected occupation measure of the discrete curve, properly renormalized by the chosen time parametrization, converges to the occupation density of the SLE(2) curve, the so-called SLE Green’s function. Although we do not prove this convergence, we rigorously establish some partial results in this direction including a new loop-erased random walk estimate.     

Conformal curves on the WO3 surface

We have studied the isoheight lines on the WO3 surface as a physical candidate for conformally invariant curves. We have shown that these lines are conformally invariant with the same statistics of domain walls in the critical Ising model. They belong to the family of conformal invariant curves called Schramm-Loewner evolution (or SLE(kappa)), with diffusivity of kappa approximately 3. This can be regarded as the first experimental observation of SLE curves. We have also argued that Ballistic Deposition (BD) can serve as a growth model giving rise to contours with similar statistics at large scales.     

On Conformally Invariant CLE Explorations

We study some conformally invariant dynamic ways to construct the Conformal Loop Ensembles with simple loops introduced in earlier papers by Sheffield, and by Sheffield and Werner. One outcome is a conformally invariant way to measure a distance of a CLE₄ loop to the boundary “within” the CLE₄, when one identifies all points of each loop.     

Free Energies and Fluctuations for the Unitary Brownian Motion

We show that the Laplace transforms of traces of words in independent unitary Brownian motions converge towards an analytic function on a non trivial disc. These results allow one to study the asymptotic behavior of Wilson loops under the unitary Yang–Mills measure on the plane with a potential. The limiting objects obtained are shown to be characterized by equations analogue to Schwinger–Dyson’s ones, named here after Makeenko and Migdal.     

A Class of Baker–Akhiezer Arrangements

We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh–Veselov Baker–Akhiezer function. These arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities. We describe all the Baker–Akhiezer arrangements when at most one line has multiplicity higher than 1. We study associated algebras of quasi-invariants which are isomorphic to the commutative algebras of quantum integrals for the generalized Calogero–Moser operators. We compute the Hilbert series of these algebras and we conclude that the algebras are Gorenstein. We also show that there are no other arrangements with Gorenstein algebras of quasi-invariants when at most one line has multiplicity bigger than 1.     

Dynamics of the Chaplygin ball on a rotating plane

This paper addresses the problem of the Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In this case, the equations of motion admit area integrals, an integral of squared angular momentum and the Jacobi integral, which is a generalization of the energy integral, and possess an invariant measure. After reduction the problem reduces to investigating a three-dimensional Poincaré map that preserves phase volume (with density defined by the invariant measure). We show that in the general case the system’s dynamics is chaotic.     

Computing the Loewner Driving Process of Random Curves in the Half Plane

We simulate several models of random curves in the half plane and numerically compute the stochastic driving processes that produce the curves through the Loewner equation. Our models include models whose scaling limit is the Schramm-Loewner evolution (SLE) and models for which it is not. We study several tests of whether the driving process is Brownian motion, as it is for SLE. We find that testing only the normality of the process at a fixed time is not effective at determining if the random curves are an SLE. Tests that involve the independence of the increments of Brownian motion are much more effective. We also study the zipper algorithm for numerically computing the driving function of a simple curve. We give an implementation of this algorithm which runs in a time O(N ^1.35) rather than the usual O(N ²), where N is the number of points on the curve.     

On variational symmetry of defect potentials and multiscale configurational force

In this work, we study invariant properties of defect potentials that are capable of describing defect motions in a continuum. By formulating two canonical defect theories, a generalized Nye theory and the Kröner–de Wit theory, we have found three defect potentials that are variational, i.e. their associated Euler–Lagrange equations are differential compatibility conditions of the continuum and defects. Consequently, symmetry properties of these variational functionals render several classes of new conservation laws and invariant integrals that are related with continuum compatibility conditions, which are independent of the constitutive relations of the continuum. The contour integral of the corresponding conserved quantity is path-independent, if the domain encompassed by such an integral is specifically defect-free. The invariant integral is applied to study macroscopically brittle fracture, and a multiscale Griffith criterion is proposed, which leads to a rigorous justification of the well-known Griffith–Irwin theory.     

Interpreting Quantum Interference Using a Berry’s Phase-like Quantity

We show that quantum interference can be interpreted in terms of a phase invariant quantity, not unlike the Berry’s phase. Under this interpretation, closed loops in time become fundamental quantum entities, and all quantum states become periodic. Decoherence is then seen to occur naturally as a consequence. This formalism, although counterintuitive, provides another useful way of assigning meaning to quantum probabilities and quasi-probabilities.     

Invariant Measures for Cherry Flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.     

Ultracold Gases and Quantum Information: Lecture Notes of the Les Houches Summer School in Singapore,edited by C. Miniatura,L-C. Kwek,M. Ducloy,B. Grémaud,B-G. Englert,L.F. Cugliandolo,A. Ekert and K. K. Phua

Measurements of the Faraday magneto-optical effect and of optical absorption in magnetic garnet crystals and films are reviewed. Earlier work was restricted to measurements in the visible spectrum; these measurements showed that the transparency was sufficient to allow magnetic domains to be studied in relatively thick slabs. Within the last year or so interest has been renewed in extending magneto-optical measurements beyond the visible spectrum. In particular, extremely low absorptions are found in the wavelength range of approximately 1–5 microns. Thie work is, in no small way, spurred by the possibility of developing a light-beam modulator using the Faraday effect in this wavelength range. Recent measurements at very short wavelengths using garnet films are also reported. By extending the short wavelength measurements to 3000 Å, about two orders of magnitude increase in Fareday rotation and optical absorption are observed, compared with the visible spectrum measurements.

The Faraday effect has been used to measure magnetic hysteresis loops in single-crystal garnet slabs. Magneto-optical observation of the corresponding domain structures shows them to be particularly simple. Detailed correlation of domain structure and hysteresis loops is described; in addition, an estimate of the domain wall energy and its width is made from these measurements.

The garnets referred to in this review are yttrium iron garnet (YIG), gadolinium iron garnet (GdIG), and YIG doped with gallium (YGaIG).     

The Volume of a Differentiable Stack

We extend the notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the volume of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the base of the groupoid. Invariant sections of a square root of this line bundle constitute an “intrinsic Hilbert space” of the stack.     

磁泡材料的磁滞迴线和泡阵磁畴的剩磁

本文采用一种用光电倍增管作为检测装置的技术,可以自动绘出透明磁性介质的磁滞迴线,测得了不同磁泡外延膜样品,在不同磁畴形状等条件下的磁迴线;解释了曲线的特点,并从实验上证实,具有泡阵磁畴的磁泡膜是处于一种特殊的剩磁状态。 关键词：     

Fractal properties of critical invariant curves

We examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers, through both the thermodynamic formalism and numerical computation. The maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed. Our results indicate that the dimension is universal for a given type of singularity and rotation number, and that among all rotation numbers, the golden mean produces the largest dimension.     

Additional Invariants and Statistical Equilibria for the 2D Euler Equations on a Spherical Domain

The role of the domain geometry for the statistical mechanics of 2D Euler flows is investigated. It is shown that for a spherical domain, there exists invariant subspaces in phase space which yield additional angular momentum, energy and enstrophy invariants. The microcanonical measure taking into account these invariants is built and a mean-field, Robert–Sommeria–Miller theory is developed in the simple case of the energy-enstrophy measure. The variational problem is solved analytically and a partial energy condensation is obtained. The thermodynamic properties of the system are also discussed.     

Uniqueness of the Invariant Measure¶for a Stochastic PDE Driven by Degenerate Noise

We consider the stochastic Ginzburg–Landau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The low-lying frequencies are then only connected to this forcing through the non-linear (cubic) term of the Ginzburg–Landau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof combine a controllability argument for thelow-lying frequencies with an infinite dimensional version of the Malliavin calculus to show positivity and regularity of the invariant measure. This then implies the uniqueness of that measure. Received: 10 September 2000 / Accepted: 13 December 2000     

Averaged number of visits

We introduce a new indicator for dynamical systems, namely the averaged number of visits, to estimate the frequency of visits in small regions when a map is iterated up to the inverse of the measure of this region. We compute this quantity analytically and numerically for various systems and we show that it depends on the ergodic properties of the systems and on their topological properties, such as the presence of periodic points.     

一种基于简正波模态消频散变换的声源距离深度估计方法

针对浅海环境中传播的低频宽带水声脉冲信号,基于简正波水平波数差和波导不变量之间的关系,本文提出了一种利用距离-频散参数二维平面聚焦测距与匹配模态能量定深的目标声源定位方法.首先,通过将由频散参数和波导不变量表示的前几阶模态相速度与由环境模型计算的相速度进行对比分析,从而估计出前几阶模态的频散参数和环境的波导不变量.其次,利用估计出的频散参数值和波导不变量对接收信号进行消频散变换处理,只有当接收信号的距离参数等于目标声源距离时,各号简正波的幅度均达到最大值,在距离-频散参数二维平面上,出现声压聚焦的现象,利用此现象可以估计目标声源的距离.不仅如此,消频散变换后的接收信号,前几阶模态在时域上明显地分离开来,可以准确地估计出前几阶模态的能量,采用多模态能量匹配的方式,可以估计出目标声源的深度.最后,通过对仿真和冬季获得的气枪信号数据处理结果验证了本文方法的有效性.     

The experimental localization of Aubry-Mather sets using regularization techniques inspired by viscosity theory

We provide a new method for the localization of Aubry-Mather sets in quasi-integrable two-dimensional twist maps. Inspired by viscosity theories, we introduce regularization techniques based on the new concept of "relative viscosity and friction," which allows one to obtain regularized parametrizations of invariant sets with irrational rotation number. Such regularized parametrizations allow one to compute a curve in the phase-space that passes near the Aubry-Mather set, and an invariant measure whose density allows one to locate the gaps on the curve. We show applications to the "golden" cantorus of the standard map as well as to a more general case.     

On noninvertible mappings of the plane: Eruptions

In this paper we are concerned with the dynamics of noninvertible transformations of the plane. Three examples are explored and possibly a new bifurcation, or "eruption," is described. A fundamental role is played by the interactions of fixed points and singular curves. Other critical elements in the phase space include periodic points and an invariant line. The dynamics along the invariant line, in two of the examples, reduces to the one-dimensional Newton's method which is conjugate to a degree two rational map. We also determine, computationally, the characteristic exponents for all of the systems. An unexpected coincidence is that the parameter range where the invariant line becomes neutrally stable, as measured by a zero Lyapunov exponent, coincides with the merging of a periodic point with a point on a singular curve. (c) 1996 American Institute of Physics.