SPECTRUM OF THE LOOP TRANSFER MATRIX ON FINITE LATTICES

We consider a model of random surfaces with extrinsic curvature term embedded into 3-D Euclidean lattice Z3. On a 3-D Euclidean lattice it has an equivalent representation in terms of the transfer matrix K(Qi, Qf), which describes the propagation of the loops Q. We study the spectrum of the transfer matrix K(Qi, Qf) on finite-dimensional lattices. The renormalisation group technique is used to investigate the phase structure of the model and its critical behaviour.

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RENORMALIZATION OF COUPLINGS IN EMBEDDED RANDOM SURFACES

Modern Physics Letters A ◽

10.1142/s0217732392003165 ◽

1992 ◽

Vol 07 (40) ◽

pp. 3747-3757

Author(s):

SUMIT R. DAS ◽

S. KALYANA RAMA

Keyword(s):

Phase Space ◽

Extra Dimension ◽

Extrinsic Curvature ◽

String Tension ◽

Length Scale ◽

Random Surfaces ◽

Curvature Term ◽

Composite Field ◽

Physical Scale ◽

The Relationship

We study the dressing of operators and flows of corresponding couplings in models of embedded random surfaces. We show that these dressings can be obtained by applying the methods of David and Distler and Kawai. We consider two extreme limits. In the first limit the string tension is large and the dynamics is dominated by the Nambu-Goto term. We analyze this theory around a classical solution in the situation where the length scale of the solution is large compared to the length scale set by the string tension. Couplings get dressed by the Liouville mode (which is now a composite field) in a non-trivial fashion. However this does not imply that the excitations around a physical “long string” have a phase space corresponding to an extra dimension. In the second limit the string tension is small and the dynamics is governed by the extrinsic curvature term. We show, perturbatively, that in this theory the relationship between the induced metric and the worldsheet metric is “renormalized,” while the extrinsic curvature term receives a non-trivial dressing as well. This has the consequence that in a generic situation the dependence of couplings on the physical scale is different from that predicted by their beta functions.

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Enumerating periodic patterns of synchrony via finite bidirectional networks

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2009.0404 ◽

2009 ◽

Vol 466 (2115) ◽

pp. 891-910

Author(s):

Ana Paula S. Dias ◽

Eliana Manuel Pinho

Keyword(s):

Nearest Neighbour ◽

Periodic Patterns ◽

Finite Dimensional ◽

Spatial Periodicity ◽

Coupled Cell Networks ◽

Euclidean Lattice ◽

Lattice Network ◽

Bidirectional Networks ◽

The Given ◽

Cell Networks

Periodic patterns of synchrony are lattice networks whose cells are coloured according to a local rule, or balanced colouring, and such that the overall system has spatial periodicity. These patterns depict the finite-dimensional flow-invariant subspaces for all the lattice dynamical systems, in the given lattice network, that exhibit those periods. Previous results relate the existence of periodic patterns of synchrony, in n -dimensional Euclidean lattice networks with nearest neighbour coupling architecture, with that of finite coupled cell networks that follow the same colouring rule and have all the couplings bidirectional. This paper addresses the relation between periodic patterns of synchrony and finite bidirectional coloured networks. Given an n -dimensional Euclidean lattice network with nearest neighbour coupling architecture, and a colouring rule with k colours, we enumerate all the periodic patterns of synchrony generated by a given finite network, or graph. This enumeration is constructive and based on the automorphisms group of the graph.

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Exact transfer-matrix enumeration and critical behaviour of self-avoiding walks across finite strips

Journal of Physics A General Physics ◽

10.1088/0305-4470/27/12/014 ◽

1994 ◽

Vol 27 (12) ◽

pp. 4055-4067 ◽

Cited By ~ 4

Author(s):

M T Batchelor ◽

C M Yung

Keyword(s):

Transfer Matrix ◽

Critical Behaviour ◽

Self Avoiding Walks

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Stiffness dependence of deconfinement transition for strings with extrinsic curvature term

Physics Letters B ◽

10.1016/0370-2693(89)91018-6 ◽

1989 ◽

Vol 225 (1-2) ◽

pp. 107-111 ◽

Cited By ~ 8

Author(s):

G. Germán ◽

H. Kleinert

Keyword(s):

Extrinsic Curvature ◽

Curvature Term

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RENORMALIZATION OF THE SINE-GORDON MODEL AND NONCONSERVATION OF THE KINK CURRENT

International Journal of Modern Physics A ◽

10.1142/s0217751x91000253 ◽

1991 ◽

Vol 06 (03) ◽

pp. 409-429 ◽

Cited By ~ 23

Author(s):

KERSON HUANG ◽

JANOS POLONYI

Keyword(s):

Renormalization Group ◽

Phase Structure ◽

Continuum Limit ◽

Renormalization Group Flow ◽

Small Coupling ◽

Vortex Density ◽

Euclidean Lattice ◽

Sine Gordon Model ◽

Group Flow ◽

The Continuum

We renormalize the (1+1)-dimensional sine-Gordon model by placing it on a Euclidean lattice, and study the renormalization group flow. We start with a compactified theory with controllable vortex activity. In the continuum limit the theory has a phase in which the kink current is anomalous, with divergence given by the vortex density. The phase structure is quite complicated. Roughly speaking, the system is normal for small coupling T. At the Kosterlitz-Thouless point T=π/2, the current can become anomalous. At the Coleman point T=8π, either the current becomes anomalous or the theory becomes trivial.

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