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Colored Tensor Models  a Review
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2012
"... Abstract. Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor mo ..."
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Cited by 40 (5 self)
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Abstract. Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating twodimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1/N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger–Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions), nontrivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.
Fatgraph models of proteins
, 2010
"... We introduce a new model of proteins that extends and enhances the traditional graphical representation by associating a combinatorial object called a fatgraph to any protein based upon its intrinsic geometry. Fatgraphs can easily be stored and manipulated as triples of permutations, and these metho ..."
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Cited by 15 (5 self)
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We introduce a new model of proteins that extends and enhances the traditional graphical representation by associating a combinatorial object called a fatgraph to any protein based upon its intrinsic geometry. Fatgraphs can easily be stored and manipulated as triples of permutations, and these methods are therefore amenable to fast computer implementation. Applications include the refinement of structural protein classifications and the prediction of geometric and other properties of proteins from their chemical structures.
Analyticity of the free energy of a closed 3manifold
 AND ASYMPTOTICS OF GRAPH COUNTING PROBLEMS IN UNORIENTED SURFACES 23
"... Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative ChernSimons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold ..."
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Cited by 12 (3 self)
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Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative ChernSimons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold is uniformly Gevrey1. As a corollary, it follows that the genus g part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of BenderGaoRichmond on the asymptotics of the number of rooted maps for arbitrary genus. We illustrate our results with an explicit formula for the free energy of a Lens space. In addition, using the Painlevé differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of BenderGaoRichmond.
On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
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Cited by 12 (1 self)
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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.
Communications in Mathematical Physics Duality of Orthogonal and Symplectic Matrix Integrals
, 2003
"... Abstract: We present an asymptotic expansion for quaternionic selfadjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their nonorientable counterparts. We show that the 2N × 2N Gaussian Orthogonal Ensemble (GOE) and N × N Gaussian Symplectic Ens ..."
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Abstract: We present an asymptotic expansion for quaternionic selfadjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their nonorientable counterparts. We show that the 2N × 2N Gaussian Orthogonal Ensemble (GOE) and N × N Gaussian Symplectic Ensemble (GSE) have exactly the same expansion term by term, except that the contributions from graphs on a nonorientable surface with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials, demonstrating that this duality is equivalent to Poincaré duality of graphs drawn on a compact surface. Another consequence of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble.
Symmetry, Integrability and Geometry: Methods and Applications Colored Tensor Models – a Review ⋆
"... Abstract. Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor mo ..."
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Abstract. Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating twodimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1/N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger–Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions), nontrivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.
ABST RACT In this thesis I review the Symmetric Unitary One Matrix Models (UMM). In the
"... this paper. I close this section by discussing the possibility of an operator formalism in the PS basis similar to the one found for UMM [12]. As I already mentioned, it is not possible to obtain the continuum limit of the action of the operators z, z \Sigma , z@ z etc. on the PS basis as the actio ..."
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this paper. I close this section by discussing the possibility of an operator formalism in the PS basis similar to the one found for UMM [12]. As I already mentioned, it is not possible to obtain the continuum limit of the action of the operators z, z \Sigma , z@ z etc. on the PS basis as the action of differential operators. The reason is that the number of non zero off diagonal lines of the matrices representing those operators 59 is infinite in the large N limit. Neuberger considered a different set of operators having a finite number of non zero off diagonal lines. The formalism is more complicated than in the case of HMM and the solutions for generic multicritical points are not known. In the next section I discuss a simpler formulation of the problem in the next section that will lead us to these solutions. In the following I omit most of the details, referring the reader to [12]. Consider the orthonormal polynomials e P n (z) = 1 p hn P n (z). From (5.1.3) we have z( e P n (z) \Gamma S n S n\Gamma1 s h n h n\Gamma1 e P n\Gamma1 (z)) = s h n+1 h n e P n+1 (z) \Gamma S n S n\Gamma1 e P n (z) : (6:1:24) Define operators A and B by z 1 X m=0 Anm e Pm (z) = 1 X m=0 Bnm e Pm (z) : (6:1:25) Then only A j j , A j j+1 B j j and B j j+1 are non zero. Consider the space spanned by the vectors / x (z) = 1 X m=0 xm e Pm (z) ; xm 2 C ; such that R d/ x (1=z)/ y (z) = P m x
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 080, 20 pages Analyticity of the Free Energy of a Closed 3Manifold ⋆
, 809
"... Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative Chern–Simons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold ..."
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Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative Chern–Simons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold is uniformly Gevrey1. As a corollary, it follows that the genus g part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of Bender–Gao–Richmond on the asymptotics of the number of rooted maps for arbitrary genus. We illustrate our results with an explicit formula for the free energy of a Lens space. In addition, using the Painlevé differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of Bender–Gao–Richmond.
Numerical Solution of Dyson Brownian Motion and a Sampling Scheme for Invariant Matrix Ensembles
"... Abstract The Dyson Brownian Motion (DBM) describes the stochastic evolution of N points on the line driven by an applied potential, a Coulombic repulsion and identical, independent Brownian forcing at each point. We use an explicit tamed Euler scheme to numerically solve the Dyson Brownian motion an ..."
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Abstract The Dyson Brownian Motion (DBM) describes the stochastic evolution of N points on the line driven by an applied potential, a Coulombic repulsion and identical, independent Brownian forcing at each point. We use an explicit tamed Euler scheme to numerically solve the Dyson Brownian motion and sample the equilibrium measure for nonquadratic potentials. The Coulomb repulsion is too singular for the SDE to satisfy the hypotheses of rigorous convergence proofs for tamed Euler schemes (Hutzenthaler et al. in Ann. Appl. Probab. 22(4):1611–1641, 2012). Nevertheless, in practice the scheme is observed to be stable for time steps of O(1/N2) and to relax exponentially fast to the equilibrium measure with a rate constant of O(1) independent of N. Further, this convergence rate appears to improve with N in accordance with O(1/N) relaxation of local statistics of the Dyson Brownian motion. This allows us to use the Dyson Brownian motion to sample N ×N Hermitian matrices from the invariant ensembles. The computational cost of generating M independent samples is O(MN4) with a naive scheme, and O(MN3 logN) when a fast multipole method is used to evaluate the Coulomb interaction.
Analyticity of the Free Energy of a Closed 3Manifold?
"... The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative Chern–Simons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold is uniform ..."
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The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative Chern–Simons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold is uniformly Gevrey1. As a corollary, it follows that the genus g part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of Bender–Gao–Richmond on the asymptotics of the number of rooted maps for arbitrary genus. We illustrate our results with an explicit formula for the free energy of a Lens space. In addition, using the Painlev´e differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of Bender–Gao–Richmond.