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865
Scale invariance of the PNG droplet and the Airy process
 J. Stat. Phys
"... We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process, A(y). The Airy process is stationary, it has continuous sample paths, its single “time ” (fixed y) distribution is the T ..."
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Cited by 93 (13 self)
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We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process, A(y). The Airy process is stationary, it has continuous sample paths, its single “time ” (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y−2. Roughly the Airy process describes the last line of Dyson’s Brownian motion model for random matrices. Our construction uses a multi–layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation. 1 The PNG droplet The polynuclear growth (PNG) model is a simplified model for layer by layer growth [1, 2]. Initially one has a perfectly flat crystal in contact with its supersaturated vapor. Once in a while a supercritical seed is formed, which then spreads laterally by further attachment of particles at its perimeter sites. Such islands coalesce if they are in the same layer and further islands may be nucleated upon already existing ones. The PNG model ignores the lateral lattice
Iterated function systems and permutation representations of the Cuntz algebra
, 1996
"... We study a class of representations of the Cuntz algebras ON, N = 2, 3,..., acting on L 2 (T) where T = R�2πZ. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the ONirreducibles decompose when rest ..."
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Cited by 74 (19 self)
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We study a class of representations of the Cuntz algebras ON, N = 2, 3,..., acting on L 2 (T) where T = R�2πZ. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the ONirreducibles decompose when restricted to the subalgebra UHFN ⊂ ON of gaugeinvariant elements; and we show that the whole structure is accounted for by arithmetic and combinatorial properties of the integers Z. We have general results on a class of representations of ON on Hilbert space H such that the generators Si as operators permute the elements in some orthonormal basis for H. We then use this to extend our results from L 2 (T) to L 2 ( T d) , d> 1; even to L 2 (T) where T is some fractal version of the torus which carries more of the algebraic
Invariant subspaces and hyperreflexivity for free semigroup algebras, preprint
 Proc. London Math. Soc. 78
, 1999
"... In this paper, we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras. This enables us to prove that they are reflexive, and moreover to obtain a quantitative measure of the distance to these algebras in term ..."
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Cited by 73 (18 self)
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In this paper, we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras. This enables us to prove that they are reflexive, and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces. Such algebras are called hyperreflexive. This property is very strong, but it has been established in only a very few cases. Moreover the prototypes of this class of algebras are the natural candidate for a noncommutative analytic Toeplitz algebra on n variables. The case we make for this analogy is very compelling. In particular, in this paper, the key to the invariant subspace analysis is a good analogue of the Beurling theorem for invariant subspaces of the unilateral shift. This leads to a notion of inner–outer factorization in these algebras. In a sequel to this paper [13], we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in C n. A free semigroup algebra is the weak operator topology closed algebra generated by a set S1,..., Sn of isometries with pairwise orthogonal ranges. These conditions are described algebraically by (F) or equivalently by
The generally covariant locality principle  A new paradigm for local quantum physics
 COMMUN.MATH.PHYS
, 2001
"... A new approach to the modelindependent description of quantum field theories will be introduced in the present work. The main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity, thus giving rise to the concept of a locally cova ..."
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Cited by 67 (13 self)
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A new approach to the modelindependent description of quantum field theories will be introduced in the present work. The main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity, thus giving rise to the concept of a locally covariant quantum field theory. Such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories, on one side, of globally hyperbolic spacetimes with isometric embeddings as morphisms and, on the other side, of ∗algebras with unital injective ∗endomorphisms as morphisms. Moreover, locally covariant quantum fields can be described in this framework as natural transformations between certain functors. The usual HaagKastler framework of nets of operatoralgebras over a fixed spacetime backgroundmanifold, together with covariant automorphic actions of the isometrygroup of the background spacetime, can be regained from this new approach as a special case. Examples of this new approach are also outlined. In case that a locally covariant quantum field theory obeys the
Operator algebras and conformal field theory  III. Fusion of positive energy representations of LSU(N) using bounded operators
, 1998
"... ..."
The Conformal spin and statistics theorem
 Commun. Math. Phys
, 1996
"... During the recent years Conformal Quantum Field Theory has become a widely studied topic, especially on a low dimensional spacetime, because of physical motivations such as the desire of a better understanding of twodimensional critical phenomena, and also for its rich mathematical structure provi ..."
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Cited by 64 (23 self)
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During the recent years Conformal Quantum Field Theory has become a widely studied topic, especially on a low dimensional spacetime, because of physical motivations such as the desire of a better understanding of twodimensional critical phenomena, and also for its rich mathematical structure providing remarkable connections with different areas such as Hopf algebras, low dimensional topology, knot invariants, subfactors
Geometric Modular Action and Spacetime Symmetry Groups
, 1998
"... A condition of geometric modular action is proposed as a selection principle for physically interesting states on general spacetimes. This condition is naturally associated with transformation groups of partially ordered sets and provides these groups with projective representations. Under suitable ..."
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Cited by 54 (9 self)
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A condition of geometric modular action is proposed as a selection principle for physically interesting states on general spacetimes. This condition is naturally associated with transformation groups of partially ordered sets and provides these groups with projective representations. Under suitable additional conditions, these groups induce groups of point transformations on these spacetimes, which may be interpreted as symmetry groups. The consequences of this condition are studied in detail in application to two concrete spacetimes – fourdimensional Minkowski and threedimensional de Sitter spaces – for which it is shown how this condition characterizes the states invariant under the respective isometry group. An intriguing new algebraic characterization of vacuum states is given. In addition, the logical relations between the condition proposed in this paper and the condition of modular covariance, widely used
Asymptotic completeness in quantum field theory. Massive PauliFierz Hamiltonians
, 1997
"... Spectral and scattering theory of massive PauliFierz Hamiltonians is studied. Asymptotic completeness of these Hamiltonians is shown. The proof consists of three parts. The first is a construction of asymptotic fields and a proof of their Fock property. The second part is a geometric analysis of ob ..."
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Cited by 54 (7 self)
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Spectral and scattering theory of massive PauliFierz Hamiltonians is studied. Asymptotic completeness of these Hamiltonians is shown. The proof consists of three parts. The first is a construction of asymptotic fields and a proof of their Fock property. The second part is a geometric analysis of observables. Its main result is what we call geometric asymptotic completeness. Finally, the last part is a proof of asymptotic completeness itself. 1 Introduction Our paper is devoted to a class of Hamiltonians used in physics to describe a quantum system ("matter" or "an atom") interacting with a bosonic field ("radiation"). K and K are respectively the Hilbert space and the Hamiltonian describing the matter. The bosonic field is described by a Fock space \Gamma(h) with the oneparticle space eg. h = L 2 (IR d ; dk), where IR d is the momentum space, and a free Hamiltonian of the form d\Gamma(!(k)) = Z !(k)a (k)a(k)dk: The function !(k) is called the dispersion relation. The inte...