Recursions in Calogero-Sutherland Model Based on Virasoro Singular Vectors

The present work is much motivated by finding an explicit way in the construction of the Jack symmetric function, which is the spectrum generating function for the Calogero-Sutherland (CS) model. To accomplish this work, the hidden Virasoro structure in the CS model is much explored. In particular, we found that the Virasoro singular vectors form a skew hierarchy in the CS model. Literally, skew is analogous to coset, but here specifically refer to the operation on the Young tableaux. In fact, based on the construction of the Virasoro singular vectors, this hierarchical structure can be used to give a complete construction of the CS states, i.e. the Jack symmetric functions, recursively. The construction is given both in operator formalism as well as in integral representation. This new integral representation for the Jack symmetric functions may shed some insights on the spectrum constructions for the other integrable systems.     

Kinetic Gaussian Model with Long-Range Interactions

In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as 1/rd δ (δ＞ 0) on d-dimensional hypercubic lattices (d = 1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all z = 2 if δ＞ 2, which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if 0 ＜δ＜ 2. It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.     

Kinetic Gaussian Model with Long-Range Interactions

In this paper dynamical critical phenomena of the Gaussian model with long-range interactions decaying as 1/r^d+δ (δ>0) on d-dimensional hypercubic lattices (d=1, 2, and 3) are studied. First, the critical points are exactly calculated, and it is found that the critical points depend on the value of δ and the range of interactions. Then the critical dynamics is considered. We calculate the time evolutions of the local magnetizations and the spin-spin correlation functions, and further the dynamic critical exponents are obtained. For one-, two- and three-dimensional lattices, it is found that the dynamic critical exponents are all z=2 if δ>2, which agrees with the result when only considering nearest neighboring interactions, and that they are all δ if 0<δ<2. It shows that the dynamic critical exponents are independent of the spatial dimensionality but depend on the value of δ.     

Stochastic Symmetry-Breaking in a Gaussian Hopfield Model

We study a two-pattern Hopfield model with Gaussian disorder. We find that there are infinitely many pure states at low temperatures in this model, and that the metastate is supported on an infinity of symmetric pairs of pure states. The origin of this phenomenon is the random breaking of a rotation symmetry of the distribution of the disorder variables.     

Harmonic Pinnacles in the Discrete Gaussian Model

The 2D Discrete Gaussian model gives each height function \({\eta : {\mathbb{Z}^2\to\mathbb{Z}}}\) a probability proportional to \({\exp(-\beta \mathcal{H}(\eta))}\), where \({\beta}\) is the inverse-temperature and \({\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2}\) sums over nearest-neighbor bonds. We consider the model at large fixed \({\beta}\), where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an \({L\times L}\) box with 0 boundary conditions concentrates on two integers M, M + 1 with \({M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}}\). The key is a large deviation estimate for the height at the origin in \({\mathbb{Z}^{2}}\), dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on \({\eta\geq 0}\) (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where \({H\sim M/\sqrt{2}}\). This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5–6):743–798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order \({\sqrt{\log L}}\). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.     

Modified Virasoro and super-Virasoro algebra

Applying the paraquantization of order Q to an open bosonic and spining strings, modified Virasoro and super-Virasoro algebra are obtained. It is shown that the anomaly c-number term is a linear function of Q.     

Quantization on the Virasoro group

The quantization of the Virasoro group is carried out by means of a previously established group approach to quantization. We explicitly work out the two-cocycles on the Virasoro group as a preliminary step. In our scheme the carrier space for all the Virasoro representations is made out of polarized functions on the group manifold. It is proved that this space does not contain null vector states, even forc1, although it is not irreducible. The full reduction is achieved in a striaghtforward way by just taking a well defined invariant subspace _{(c, h)} , the orbit of the enveloping algebra through the vacuum, which is irreducible for any value ofc andh. _{(c, h)} is a proper subspace of the space of polarized functions for those values ofc andh for which the Kac determinant is zero. We give the local version of these group representations as well as the associated classical phase space structures, i.e., symplectic form and Noether invariants.Research partially supported by the Conselleria de Cultura de la Generalitat Valenciana, the Plan de formación del Personal investigador, the Comision Interministerial de Ciencia y Tecnologia (CICYT) and the British Council     

The Gaussian Model on the Inhomogeneous Fractal Lattices

The decimation real-space renormalization group and spin-rescaling methods are applied to the study of phase transition of the Gaussian model on fractal lattices. It is found that the critical point K^* equals b/2 ( b is the distribution constant of Gaussian model) on nonbranching Koch curves. For inhomogeneous fractal lattices, it is proposed that the b is replaced with b_qi (qi is the coordination number of the site i) and satisfies a certain relation b_qi/b_qj = q_i/q_j. Under this supposition we find that the critical point of the Gaussian model on a branching Koch curve can be expressed uniquely as K^* = b_qi/q_i.     

离焦写入线宽的动态高斯模型

为提高集成光学器件中的多线宽制作效率,提出了基于激光直接写入技术的离焦加工模式,即加工时激光束的聚焦点不落在胶层表面而是在其前方或后方;同时还建立了基于多个加工参量的线宽数学模型。此线宽模型首先假定激光束经物镜变换后,在像方任意离焦面上的能量仍保持高斯分布,其次还考虑了光束动态扫描引起的光斑线度对曝光能量分布的影响,故称其为动态高斯模型。此线宽模型涉及光功率、基片离焦量、光束扫描速度、胶层曝光能量阈值等加工参量。对线宽模型的验证在自行研发的极坐标型激光图形发生器上进行,实验表明:相对于不考虑光斑运动的静态高斯模型,动态高斯模型更为符合实验结果。     

A Gaussian Model for Anisotropic Strange Quark Stars

For studying the anisotropic strange quark stars,we assume that the radial pressure inside an anisotropic star can be obtained simply by isotropic pressure plus an additional Gaussian term with three free parameters(A,μand χ).According to recent observations,a pulsar in a mass range of 1.97±0.04M_⊙ has been measured.Hence,we take this opportunity to set the free parameters of our model.We fix χ by applying boundary and stability conditions and then search the A —μ parameter space for a maximum mass range of 1.9M_⊙ M_(max) 2.1M_⊙.Our results indicate that anisotropy increases the maximum mass M_(max) and also its corresponding radius R for a typical strange quark star.Furthermore,our model shows magnetic field and electric charge increase the anisotropy factor △.In fact,△ has a maximum on the surface and this maximum goes up in the presence of magnetic field and electric charge.Finally,we show that anisotropy can be more effective than either magnetic field or electric charge in raising maximum mass of strange quari stars.     

Vertex operators for deformed Virasoro algebra

Vertex operators for the deformed Virasoro algebra are defined, their bosonic representation is constructed, and a difference equation for the simplest vertex operators is described. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 11, 876–881 (10 June 1996) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Commitment Versus Persuasion in the Three-Party Constrained Voter Model

In the framework of the three-party constrained voter model, where voters of two radical parties (A and B) interact with “centrists” (C and C _ζ ), we study the competition between a persuasive majority and a committed minority. In this model, A’s and B’s are incompatible voters that can convince centrists or be swayed by them. Here, radical voters are more persuasive than centrists, whose sub-population comprises susceptible agents C and a fraction ζ of centrist zealots C _ζ . Whereas C’s may adopt the opinions A and B with respective rates 1+δ _A and 1+δ _B (with δ _A ≥δ _B >0), C _ζ ’s are committed individuals that always remain centrists. Furthermore, A and B voters can become (susceptible) centrists C with a rate 1. The resulting competition between commitment and persuasion is studied in the mean field limit and for a finite population on a complete graph. At mean field level, there is a continuous transition from a coexistence phase when ζ<Δ _c =δ _A /(1+δ _A ) to a phase where centrism prevails when ζ≥Δ _c . In a finite population of size N, demographic fluctuations lead to centrism consensus and the dynamics is characterized by the mean consensus time τ. Because of the competition between commitment and persuasion, here consensus is reached much slower (ζ<Δ _c ) or faster (ζ≥Δ _c ) than in the absence of zealots (when τ～N). In fact, when ζ<Δ _c and there is an initial minority of centrists, the mean consensus time grows as τ～N ^?1/2 e ^Nγ , with N?1 and . The dynamics is thus characterized by a metastable state where the most persuasive voters and centrists coexist when δ _A >δ _B , whereas all species coexist when δ _A =δ _B . When ζ≥Δ _c and the initial density of centrists is low, one finds τ～lnN (when N?1). Our analytical findings are corroborated by stochastic simulations.     

Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket

The classical Volterra model, equipped with the Faddeev-Takhtajan Poisson bracket provides a lattice version of the Virasoro algebra. The Volterra model being integrable, we can express the dynamical variables in terms of the so-called separated variables. Taking the continuum limit of these formulae, we obtain the Virasoro generators written as determinants of infinite matrices, the elements of which are constructed with a set of points lying on an infinite genus Riemann surface. The coordinates of these points are separated variables for an infinite set of Poisson commuting quantities including L ₀. The scaling limit of the eigenvector can also be calculated explicitly, so that the associated Schroedinger equation is in fact exactly solvable.     

Relating Kac-Moody,Virasoro and Krichever-Novikov algebras

We demonstrate that the Kac-Moody and Virasoro-like algebras on Riemann surfaces of arbitrary genus with two punctures introduced by Krichever and Novikov are in two ways linearly related to Kac-Moody and Virasoro algebras onS ¹. The two relations differ by a Bogoliubov transformation, and we discuss the connection with the operator formalism.     

On the Representation Theory of Virasoro Nets

We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the c=1 Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge c is in a certain subset of (1, ), including [2, ), and h(c–1)/24, the irreducible representation with lowest weight h of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge c is in the above set and satisfies c25 then the corresponding Virasoro net has no proper local extensions of compact type.Supported in part by the Italian MIUR and GNAMPA-INDAM.