Solvable models of temporally correlated random walk on a lattice

We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distributionW(n, t) specifying the probability ofn jumps or steps occurring in timet. Uncorrelated diffusion occurs whenW is a Poisson distribution. The solutions corresponding to two different families of distributionsW are found and discussed. The Poissonian is a limiting case in each of these families. This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences. In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability densityψ(t) for the time interval between successive jumps.W is constructed in terms ofψ using the continuous-time random walk approach. The theory is then specialized to the case whenψ belongs to the class of special Erlangian density functions. In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated). Various aspects such as the continuum limit, the master equation forP, the asymptotic behaviour ofP, etc., are discussed.     

Differential Geometry from Differential Equations

We first show how, from the general 3rd order ODE of the form , one can construct a natural Lorentzian conformal metric on the four-dimensional space . When the function satisfies a special differential condition the conformal metric possesses a conformal Killing field, , which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space ) or equivalently, on the space of solutions of the original differential equation. This construction is then generalized to the pair of differential equations, z _ss =S(z,z _s ,z _t ,z _st ,s,t) and z _tt =T(z,z _s ,z _t ,z _st ,s,t), with z _s and z _t the derivatives of z with respect to s and t. In this case, from S and T, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,z _s ,z _t ,z _st ,s,t). When the S and T satisfy differential conditions analogous to those of the 3^rd order ode, the 6-space then possesses a pair of conformal Killing fields, and which allows, via the mapping to the four-space of (z,z _s ,z _t ,z _st ) and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions on S and T, thus includes all (local) solutions of the Einstein equations. Received: 10 October 2000 / Accepted: 26 June 2001     

A Functional-Analytic Theory¶of Vertex (Operator) Algebras, I

This paper is the first in a series of papers developing a functional-analytic theory of vertex (operator) algebras and their representations. For an arbitrary ℤ-graded finitely-generated vertex algebra (V, Y, 1) satisfying the standard grading-restriction axioms, a locally convex topological completion H of V is constructed. By the geometric interpretation of vertex (operator) algebras, there is a canonical linear map from $V⊗V to (the algebraic completion of V) realizing linearly the conformal equivalence class of a genus-zero Riemann surface with analytically parametrized boundary obtained by deleting two ordered disjoint disks from the unit disk and by giving the obvious parametrizations to the boundary components. We extend such a linear map to a linear map from $H\tilde{\otimes} H$ ( being the completed tensor product) to H, and prove the continuity of the extension. For any finitely-generated ℂ-graded V-module (W, Y _W ) satisfying the standard grading-restriction axioms, the same method also gives a topological completion H _W of W and gives the continuous extensions from to H _W of the linear maps from to realizing linearly the above conformal equivalence classes of the genus-zero Riemann surfaces with analytically parametrized boundaries. Received: 15 August 1998 / Accepted: 13 January 1999     

Spectral imaging system for non-contact colour measurement

This paper describes the development of a non-contact system for measuring colour of printed material at web speeds. The system proposed uses a non-contact spectrophotometer based on a holographic grating, in conjunction with a conventional monochrome area scan camera, from which colour spectral data is extracted, whilst a xenon flash is used to illuminate colour samples. Software and hardware details of the system are given, along with the underlying mathematics for colour space conversion and measurement. Conversion equations from X, Y, Z chromaticity co-ordinates to the RGB system are presented, and also equations to convert from the L^*a^*b^* colour space to X, Y, Z chromaticity co-ordinates. Experimental results are presented whereby the non-contact spectral system is shown to perform to a colour tolerance exceeding that of conventional colour video systems.     

Modular Categories and Orbifold Models

In this paper, we try to answer the following question: given a modular tensor category ? with an action of a compact group G, is it possible to describe in a suitable sense the “quotient” category ?/G? We give a full answer in the case when ?=?ℯ? is the category of vector spaces; in this case, ?ℯ?/G turns out to be the category of representation of Drinfeld's double D(G). This should be considered as the category theory analog of the topological identity {pt}/G=BG. This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if ? is a vertex operator algebra which has a unique irreducible module, ? itself, and G is a compact group of automorphisms of ?, and some not too restrictive technical conditions are satisfied, then G is finite, and the category of representations of the algebra of invariants, ? ^G , is equivalent as a tensor category to the category of representations of Drinfeld's double D(G). We also get some partial results in the non-holomorphic case, i.e. when ? has more than one simple module. Received: 27 August 2001 / Accepted: 1 March 2002     

Relative compatibility in conventional quantum mechanics

The notion of relative compability is introduced, according to which compatibility is construed as relative to individual quantum states. The compatibility domain of two observablesA, B is defined to be the set com(A, B) of states relative to whichA andB are compatible. Three basic categories of relative compatibility are then defined according to the character of com(A, B): absolute compatibility (ordinary compatibility), absolute incompatibility, and partial compatibility. Then com(A, B) is seen to be a subspace of Hilbert space invariant underA andB; being a subspace, it corresponds to a quantum binary observable (projection), denotedc(A, B). IfA andB are themselves binary observables, thenc(A, B) is expressible as a quantum logical compound ofA andB using the lattice operations meet, join, and orthocomplement. This suggests extending the notion of relative compatibility to the theory of orthomodular lattices, as well as to more general lattice-theoretic formulations of quantum mechanics.     

Symplectic Structures of Moduli Space¶of Higgs Bundles over a Curve and Hilbert Scheme¶of Points on the Canonical Bundle

The moduli space of triples of the form (E,θ,s) are considered, where (E,θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y,L) is the spectral data for the Higgs bundle (E,θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K _X , since Y is a curve on K _X . The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E,θ,s) to the divisor of the corresponding section of the line bundle on the spectral curve. Received: 15 January 2000 / Accepted: 25 March 2001     

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e ^πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N > 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.     

Drinfeld-Sokolov gravity

A lagrangian euclidean model of Drinfeld-Sokolov (DS) reduction leading to generalW-algebras on a Riemann surface of any genus is presented. The background geometry is given by the DS principal bundleK associated to a complex Lie groupG and anSL(2,) subgroupS. The basic fields are a hermitian fiber metricH ofK and a (0, 1) Koszul gauge fieldA ^* ofK valued in a certain negative graded subalgebrar ofg related tos. The action governing theH andA ^* dynamics is the effective action of a DS field theory in the geometric background specified byH andA ^*. Quantization ofH andA ^* implements on one hand the DS reduction and on the other defines a novel model of 2d gravity, DS gravity. The gauge fixing of the DS gauge symmetry yields an integration on a moduli space of DS gauge equivalence classes ofA ^* configurations, the DS moduli space. The model has a residual gauge symmetry associated to the DS gauge transformations leaving a given fieldA ^* invariant. This is the DS counterpart of conformal symmetry. Conformal invariance and certain non-perturbative features of the model are discussed in detail.     

Expanded Vandermonde Powers and Sum Rules for the Two-Dimensional One-Component Plasma

The two-dimensional one-component plasma (2dOCP) is a system of N mobile particles of the same charge q on a surface with a neutralizing background. The Boltzmann factor of the 2dOCP at temperature T can be expressed as a Vandermonde determinant to the power Γ=q ²/(k _B T). Recent advances in the theory of symmetric and anti-symmetric Jack polynomials provide an efficient way to expand this power of the Vandermonde in their monomial basis, allowing the computation of several thermodynamic and structural properties of the 2dOCP for N values up to 14 and Γ equal to 4, 6 and 8. In this work we explore two applications of this formalism, to the study of the pair correlation function of the 2dOCP on the sphere, and the distribution of radial statistics of the 2dOCP in the plane. Also provided is a finite N approximation to the pair correlation on the sphere, and a sum rule for the constant term in the large N expansion of the moments of the density in the plane.     

Fields and connections in a fiber bundle over spacetime arising from the reduction of the multidimensional linear connection

We consider the invariant linear connections in a multidimensional universe where a compact connected Lie groupG acts with a single orbit typeG/H. It is shown that every such connection reduces to a principal connection in a dimensionally-reduced fiber bundle over the four-dimensional spacetimeM and to a set of fields onM with a gauge group isomorphic toN/H × GL(4,R), whereN is the normalizer inG of the closed subgroupH.     

Necessity and sufficiency of f being a divergence for vanishing of variational derivatives of f: General case

Concise proofs are presented for the necessity and sufficiency of f being a divergence for the variational derivatives of f to vanish identically, where f is a function of N functions of n variables, their partial derivatives to arbitrary order, and the n variables. The approach is conventional.     

Universality of the REM for Dynamics of Mean-Field Spin Glasses

We consider a version of Glauber dynamics for a p-spin Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ _{β ,p}  > 0, such that for all exponential time scales, exp(γ N), with γ < γ _{β ,p} , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β ² < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud’s REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.     

Survival of grand unified monopoles

If a grand unified theory with a compact group G is spontaneously broken to H, magnetic monopoles are created. The fate of such an H-monopole under a subsequent breaking to K⊂H is shown to depend on the behaviour of its non-Abelian charge Q introduced by Goddard, Nuyts, and Olive: if Q belongs to the Lie algebra k of K, the monopole survives: if Q can be H-rotated to k, it can be converted. A necessary condition for an H-monopole to survive is that its Higgs charges satisfy a topological constraint.     

Smoothening transition of a two-dimensional pressurized polymer ring

We revisit the problem of a two-dimensional polymer ring subject to an inflating pressure differential. The ring is modeled as a freely jointed closed chain of N monomers. Using a Flory argument, mean-field calculation and Monte Carlo simulations, we show that at a critical pressure, p_c ∼ N^-1, the ring undergoes a second-order phase transition from a crumpled, random-walk state, where its mean area scales as 〈A〉 ∼ N, to a smooth state with 〈A〉 ∼ N². The transition belongs to the mean-field universality class. At the critical point a new state of polymer statistics is found, in which 〈A〉 ∼ N^3/2. For p ≫ p_c we use a transfer-matrix calculation to derive exact expressions for the properties of the smooth state.     

Ubiquitousness of link-density and link-pattern communities in real-world networks

Many networks are characterized by the presence of communities, densely intra-connected groups with sparser inter-connections between groups. We propose a community overlay network representation to capture large-scale properties of communities. A community overlay G _o can be constructed upon a network G, called the underlying network, by (a) aggregating each community in G as a node in the overlay G _o ; (b) connecting two nodes in the overlay if the corresponding two communities in the underlying network have a number of direct links in between, (c) assigning to each node/link in the overlay a node/link weight, which represents e.g. the percentage of links in/between the corresponding underlying communities. The community overlays have been constructed upon a large number of real-world networks based on communities detected via five algorithms. Surprisingly, we find the following seemingly universal properties: (i) an overlay has a smaller degree-degree correlation than its underlying network ρ _o (D _l+, D _l−) < ρ(D _l+, D _l−) and is mostly disassortative ρ _o (D _l+, D _l−) < 0; (ii) a community containing a large number W _i of nodes tends to connect to many other communities ρ _o (W _i , D _i ) > 0. We explain the generic observation (i) by two facts: (1) degree-degree correlation or assortativity tends to be positively correlated with modularity; (2) by aggregating each community as a node, the modularity in the overlay is reduced and so is the assortativity. The observation (i) implies that the assortativity of a network depends on the aggregation level of the network representation, which is illustrated by the Internet topology at router and AS level.     

Numerical study of local and global persistence in directed percolation

The local persistence probability P _l (t) that a site never becomes active up to time t, and the global persistence probability P _g (t) that the deviation of the global density from its mean value does not change its sign up to time t are studied in a (1+1)-dimensional directed percolation process by Monte-Carlo simulations. At criticality, starting from random initial conditions, P _l (t) decays algebraically with the exponent . The value is found to be independent of the initial density and the microscopic details of the dynamics, suggesting is an universal exponent. The global persistence exponent is found to be equal or larger than . This contrasts with previously known cases where . It is shown that in the special case of directed-bond percolation, P _l (t) can be related to a certain return probability of a directed percolation process with an active source (wet wall). Received: 15 December 1997 / Revised: 6 April 1998 / Accepted: 29 May 1998     

On the Characteristic Polynomial¶ of a Random Unitary Matrix

We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N×N unitary matrix, as N→∞. First we show that , evaluated at a finite set of distinct points, is asymptotically a collection of i.i.d. complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function. Next we obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting Gaussian measure. Large deviations results for ln Z/A, evaluated at a finite set of distinct points, can be obtained for . For higher-order scalings we obtain large deviations results for ln Z/A evaluated at a single point. There is a phase transition at A= ln N (which only applies to negative deviations of the real part) reflecting a switch from global to local conspiracy. Received: 27 June 2000 / Accepted: 30 January 2001     

Quasi-Stationary Regime of a Branching Random Walk in Presence of an Absorbing Wall

A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as the velocity v of the wall varies. Below the critical velocity v _c , the population has a non-zero survival probability and when the population survives its size grows exponentially. We investigate the histories of the population conditioned on having a single survivor at some final time T. We study the quasi-stationary regime for v<v _c when T is large. To do so, one can construct a modified stochastic process which is equivalent to the original process conditioned on having a single survivor at final time T. We then use this construction to show that the properties of the quasi-stationary regime are universal when v→v _c . We also solve exactly a simple version of the problem, the exponential model, for which the study of the quasi-stationary regime can be reduced to the analysis of a single one-dimensional map.     

Bipartite Matching and Van der Waerden Conjecture

In this paper I show that the free energy F and the cost C associated to a bipartite matching problem can be explicitly estimated in term of the solution of a suitable system of equations (cavity equations in the following). The proof of these results relies on a well known result in combinatorics: the Van der Waerden conjecture (Egorychev–Falikman Theorem). Cavity equations, derived by a mean field argument by Mèzard and Parisi, can be considered as a smoothed form of the dual formulation for the bipartite matching problem. Moreover cavity equation are the Euler–Lagrange equations of a convex functional G parameterized by the temperature T. In term of their unique solution it is possible to define a free-energy-like function of the temperature g(T). g is a strictly decreasing concave function of T and C=g(0). The convexity of G allows to define an explicit algorithm to find the solution of the cavity equations at a given temperature T. Moreover, once the solution of the cavity equations at a given temperature T is known, the properties of g allow to find exact estimates from below and from above of the cost C.