Random Unitary Matrices, Permutations and Painlevé

This paper is concerned with certain connections between the ensemble of n×n unitary matrices – specifically the characteristic function of the random variable tr(U) – and combinatorics – specifically Ulam's problem concerning the distribution of the length of the longest increasing subsequence in permutation groups – and the appearance of Painlevé functions in the answers to apparently unrelated questions. Among the results is a representation in terms of a Painlevé V function for the characteristic function of tr(U) and (using recent results of Baik, Deift and Johansson) an expression in terms of a Painlevé II function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups. Received: 2 December 1998 / Accepted: 12 May 1999     

Experimental investigation of ligand effects on the conversion electron spectrum of the 22.5 keV M1 + E2 nuclear transition in <Superscript>149</Superscript>Sm

The conversion electron spectrum of the 22.5keV M1 + E2 nuclear transition in ¹⁴⁹Sm from the electron capture decay of ¹⁴⁹Eu was experimentally studied for the “Eu₂O₃” and “EuF₃” compounds in which ¹⁴⁹Eu ions have the same assumed oxidation number +3 . While the energies of the L, M, N, O, and P₁ conversion lines for “EuF₃” were lower, on average, by 1.7(1), 2.7(2), 2.3(3), 4.1(2), and 5.7(9)eV, respectively, than those for “Eu₂O₃”, no changes between the two ligand complexes were observed for relative intensities of the conversion lines and their natural widths within the error limits.     

A Fast Algorithm for Wave Propagation from a Plane or a Cylindrical Surface

The newly developed Taylor-Interpolation-FFT (TI-FFT) algorithm dramatically increases the computational speeds for millimeter wave propagation from a planar (cylindrical) surface onto a “quasi-planar” (“quasi-cylindrical”) surface. Two different scenarios are considered in this article: the planar TI-FFT is for the computation of the wave propagation from a plane onto a “quasi-planar” surface and the cylindrical TI-FFT is for the computation of wave propagation from a cylindrical surface onto a “quasi-cylindrical” surface. Due to the use of the FFT, the TI-FFT algorithm has a computational complexity of O(N ² log₂  N ²) for an N × N computational grid, instead of N ⁴ for the direct integration method. The TI-FFT algorithm has a low sampling rate according to the Nyquist sampling theorem. The algorithm has accuracy down to −80 dB and it works particularly well for narrow-band fields and “quasi-planar” (“quasi-cylindrical”) surfaces.     

Super Universality of the Quantum Hall Effect and the?“Large N Picture” of the ? Angle

It is shown that the “massless chiral edge excitations” are an integral and universal aspect of the low energy dynamics of the ϑ vacuum that has historically gone unnoticed. Within the SU(M+N)/S(U(M)×U(N)) non-linear sigma model we introduce an effective theory of “edge excitations” that fundamentally explains the quantum Hall effect. In sharp contrast to the common beliefs in the field our results indicate that this macroscopic quantization phenomenon is, in fact, a super universal strong coupling feature of the ϑ angle with the replica limit M=N=0 only playing a role of secondary importance. To demonstrate super universality we revisit the large N expansion of the CP ^N−1 model. We obtain, for the first time, explicit scaling results for the quantum Hall effect including quantum criticality of the quantum Hall plateau transition. Consequently a scaling diagram is obtained describing the cross-over between the weak coupling “instanton phase” and the strong coupling “quantum Hall phase” of the large N theory. Our results are in accordance with the “instanton picture” of the ϑ angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical “large N picture.”     

Cluster formation, pressure and density measurements in a granular medium fluidized by vibrations

We report experimental results on the behavior of an ensemble of inelastically colliding particles, excited by a vibrated piston in a vertical cylinder. When the particle number is increased, we observe a transition from a regime where the particles have erratic motions (“granular gas”) to a collective behavior where all the particles bounce like a nearly solid body. In the gas-like regime, we measure the density of particles as a function of the altitude and the pressure as a function of the number N of particles. The atmosphere is found to be exponential far enough from the piston, and the “granular temperature”, T, dependence on the piston velocity, V, is of the form , where is a decreasing function of N. This may explain previous conflicting numerical results. Received 1 February 1999     

On the Large Time Asymptotics¶of Decaying Burgers Turbulence

The decay of Burgers turbulence with compactly supported Gaussian “white noise” initial conditions is studied in the limit of vanishing viscosity and large time. Probability distribution functions and moments for both velocities and velocity differences are computed exactly, together with the “time-like” structure functions T _n (t,τ)≡< (u(t+τ) -u(t)) ⁿ >. The analysis of the answers reveals both well known features of Burgers turbulence, such as the presence of dissipative anomaly, the extreme anomalous scaling of the velocity structure functions and self similarity of the statistics of the velocity field, and new features such as the extreme anomalous scaling of the “time-like” structure functions and the non-existence of a global inertial scale due to multiscaling of the Burgers velocity field. We also observe that all the results can be recovered using the one point probability distribution function of the shock strength and discuss the implications of this fact for Burgers turbulence in general. Received: 4 October 1999 / Accepted: 4 February 2000     

A Characteristic Decay Semigroup for the Resonances of Trace Class Perturbations with Analyticity Conditions of Semibounded Hamiltonians

To asymptotic complete scattering systems {M ₊+V,M ₊} on H₊:=L²(R₊,K{\mathcal{H}}_{+}:=L^{2}(\mathbf{R}_{+},{\mathcal{K}}, d λ), where M ₊ is the multiplication operator on H₊{\mathcal{H}}_{+} and V is a trace class operator with analyticity conditions, a decay semigroup is associated such that the spectrum of the generator of this semigroup coincides with the set of all resonances (poles of the analytic continuation of the scattering matrix into the lower half plane across the positive half line), i.e. the decay semigroup yields a “time-dependent” characterization of the resonances. As a counterpart a “spectral characterization” is mentioned which is due to the “eigenvalue-like” properties of resonances.     

Dual Forms on Supermanifolds and Cartan Calculus

We introduce and study the complex of “stable forms” on supermanifolds. Stable forms on a supermanifold M are represented by Lagrangians of “copaths” (formal systems of equations, which may or may not specify actual surfaces) on M×ℝ ^D . Changes of D give rise to stability isomorphisms. The resulting (direct limit) {Cartan-de Rham} complex made of stable forms extends both in positive and negative degree. Its positive half is isomorphic to the complex of forms defined as Lagrangians of paths, studied earlier. Including the negative half is crucial, in particular, for homotopy invariance. For stable forms we introduce (non-obvious) analogs of exterior multiplication by covectors and contraction with vectors and find the anticommutation relations that they obey. Remarkably, the version of the Clifford algebra so obtained is based on the super anticommutators rather than the commutators and (before stabilization) it includes some central element σ. An analog of Cartan's homotopy identity is proved, which also contains this “stability operator”σ. Received: 3 January 2000 / Accepted: 15 September 2001     

Random patterns generated by random permutations of natural numbers

We survey recent results on some one- and two-dimensional patterns generated by random permutations of natural numbers. In the first part, we discuss properties of random walks, evolving on a one-dimensional regular lattice in discrete time n, whose moves to the right or to the left are prescribed by the rise-and-descent sequence associated with a given random permutation. We determine exactly the probability of finding the trajectory of such a permutation-generated random walk at site X at time n, obtain the probability measure of different excursions and define the asymptotic distribution of the number of “U-turns" of the trajectories - permutation “peaks" and “through". In the second part, we focus on some statistical properties of surfaces obtained by randomly placing natural numbers 1,2,3, ...,L on sites of a 1d or 2d lattices containing L sites. We calculate the distribution function of the number of local “peaks" - sites the number at which is larger than the numbers appearing at nearest-neighboring sites - and discuss surprising collective behavior emerging in this model.     

The mystery of the alkali metals; the induced anomalous Hall effect in thin Cs films

Sandwiches made from Fe and Cs films are investigated as a function of the magnetic field and the Cs thickness. Conduction electrons which cross from the Fe to the Cs are marked by a drift velocity component perpendicular to the electric field. The anomalous Hall effect in the Fe provides this “non-diagonal” kick to the electrons that cross from the Fe into the Cs. The ballistic propagation of the conduction electrons can be monitored as a function of the Cs film thickness. The free propagation into the Cs is measured in terms of the non-diagonal conductance L_xy which we denote as the “induced anomalous Hall conductance”L _xy ⁰. For a normal (non-magnetic) metal in contact with Fe, L_xy increases with the thickness of the normal metal until the film thickness exceeds (half) the mean free path of the conduction electrons. For Cs on top of Fe the induced anomalous Hall conductance increases up to a Cs coverage of about 100 A, then, in contrast to other non-magnetic metals, L _xy ⁰ decreases for larger Cs coverage and approaches zero. This behavior cannot be explained with the free electron model. The strange behavior of the induced AHC in Cs films adds an even more challenging mystery to the already poorly understood properties of thin Cs films. These results defy explanation in the free electron model. Received 29 April 1999 and Received in final form 10 July 1999     

Automorphisms and symmetries of quantum logics

Given an amalgam of groups then every quantum logicQ ₀ = (L ₀,M ₀) (L ₀ is aσ-orthomodular poset,M ₀ is a full set of states on it) satisfying some reasonable conditions can be embedded in a quantum logicQ = (L, M), in which (1) all the automorphisms ofL form a group ∼-G ₁, (2) all the automorphisms ofM form a group ∼-G ₂, and (3) all the symmetries ofQ form a group ∼-G ₀. The quantum logic of all closed subspaces of a Hilbert spaceH and all its measures satisfies the conditions required fromQ ₀; hence, enlarging it, one can obtain “anything.”     

Experimental study of the energy gap in the spin-wave spectrum in Co/Pd multilayer films

The energy gap appearing in the spin-wave spectrum as a result of Bragg scattering by the modulation period q=2π/(d ₁+d ₂) of a one-dimensional superlattice is observed by the method of spin-wave resonance in Co/Pd multilayer films. It is shown that this gap is asymmetric: The “positive” deviation is from two to three times greater than the “negative” deviation. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 12, 938–941 (25 June 1996)     

Classification of the Five-Dimensional Lie Superalgebras Over the Real Numbers

The purpose of this contribution, is to initiate a classification of Lie superalgebras (LS) of dimension five, over the base field ℝ of real numbers. We use the “graded skew-symmetry” and the “graded Jacobi identity” in order to get restrictions for the commutators and anticommutators of an arbitrary five-dimensional Lie superalgebra L = L ₀⊕ L ₁ PACS 2003: 02.20.Sv     

Creating a monopole in 4D gauge theories

The problem of defining the second quantized monopole creation operator in non-Abelian gauge theories is discussed and exemplified by the (3 + 1)-dimensional Georgi-Glashow model. We construct the “coherent state” operator M(x) that creates the Coulomb magnetic field in terms of the Dirac singular electromagnetic potential. Our calculation of the vacuum expectation value of this operator 〈M(x)〉 in the confining phase indicates that it is free from the singularity along the Dirac string and in the leading order of perturbation theory the 〈M(x)〉 vanishes as a power of the volume of the system. This supports the conception that inclusion of the nonperturbative effects introduces an effective infrared cutoff on the calculation providing the finiteness of vacuum expectation value 〈M(x)〉. The text was submitted by the authors in English.     

μ+SR studies on superconducting YBa2(Cu1−xFex)3Oy system

Spin-glass like magnetic ordering of iron moments was observed in both orthorhombic and tetragonal YBa₂(Cu_1−xFe_x)₃O_y (x=0.08) by μ⁺SR measurements. In a “Tetra” sample, all the muons sense the superconducting transition at 60 K and magnetic ordering at around 15 K, while in an “Ortho” sample they reveal that two magnetically different parts exist in the sample: about 40% of the sample is superconducting withT _c ≈90K and the remaining part is magnetic withT _M≈33K. These phenomena can be explained in terms of clustering of the Fe atoms in the “Ortho” sample.     

Poincaré–Lelong Approach to Universality and Scaling of Correlations Between Zeros

This note is concerned with the scaling limit as N→∞ of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers L ^N of any positive holomorphic line bundle L over a compact K?hler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more generally, of the “correlation forms” are universal, i.e. independent of the bundle L, manifold M or point on M. Received: 17 March 1999 / Accepted: 5 August 1999     

Effect of regimes of cooling in a magnetic field on the magnetization of a La0.9Sr0.1MnO3 single crystal

Magnetization measurements were performed on a lanthanum manganite La_0.9Sr_0.1MnO₃ single crystal in the temperature interval 4.2–300 K and magnetic field interval 50 Oe-55 kOe in two sample cooling regimes: 1) cooling down to 4.2 K in a high (55 kOe) magnetic field, and 2) cooling in a “zero” field. It is shown that the temperature dependences of the magnetization M(T) are substantially different in these regimes. Pronounced anomalies of M(T) were observed at temperatures T*=103 K and T _c =145 K. The first anomaly is attributed to a structural transition, while the second one corresponds to a ferromagnet-paramagnet phase transition. The magnetization of a La_0.9Sr_0.1MnO₃ single crystal in the cooling regimes studied shows typical “spin-glass” behavior. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 1, 39–43 (10 July 1998)     

Random Walks and Polymers in the Presence of Quenched Disorder

After a general introduction to the field, we describe some recent results concerning disorder effects on both ‘random walk models’, where the random walk is a dynamical process generated by local transition rules, and on ‘polymer models’, where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points: thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures T _c(i,L) over the ensemble of samples (i) of size L. We describe the results of this analysis for the bidimensional wetting and for the Poland–Scheraga model of DNA denaturation.Conference Proceedings “Mathematics and Physics”, I.H.E.S., France, November 2005