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Continuum Limit (continuum + limit)

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Continuum limits for classical sequential growth models

RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2010
Graham Brightwell
Abstract A random graph order, also known as a transitive percolation process, is defined by taking a random graph on the vertex set {0,,,n , 1} and putting i below j if there is a path i = i1,ik = j in the graph with i1 < , < ik. Rideout and Sorkin Phys. Rev. D 63 (2001) 104011 provide computational evidence that suitably normalized sequences of random graph orders have a "continuum limit." We confirm that this is the case and show that the continuum limit is always a semiorder. Transitive percolation processes are a special case of a more general class called classical sequential growth models. We give a number of results describing the large-scale structure of a general classical sequential growth model. We show that for any sufficiently large n, and any classical sequential growth model, there is a semiorder S on {0,,,n - 1} such that the random partial order on {0,,,n - 1} generated according to the model differs from S on an arbitrarily small proportion of pairs. We also show that, if any sequence of classical sequential growth models has a continuum limit, then this limit is (essentially) a semiorder. We give some examples of continuum limits that can occur. Classical sequential growth models were introduced as the only models satisfying certain properties making them suitable as discrete models for spacetime. Our results indicate that this class of models does not contain any that are good approximations to Minkowski space in any dimension , 2. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010 [source]

Field theory on a non-commutative plane: a non-perturbative study

FORTSCHRITTE DER PHYSIK/PROGRESS OF PHYSICS, Issue 5 2004
F. Hofheinz
Abstract The 2d gauge theory on the lattice is equivalent to the twisted Eguchi,Kawai model, which we simulated at N ranging from 25 to 515. We observe a clear large N scaling for the 1- and 2-point function of Wilson loops, as well as the 2-point function of Polyakov lines. The 2-point functions agree with a universal wave function renormalization. The large N double scaling limit corresponds to the continuum limit of non-commutative gauge theory, so the observed large N scaling demonstrates the non-perturbative renormalizability of this non-commutative field theory. The area law for the Wilson loops holds at small physical area as in commutative 2d planar gauge theory, but at large areas we find an oscillating behavior instead. In that regime the phase of the Wilson loop grows linearly with the area. This agrees with the Aharonov-Bohm effect in the presence of a constant magnetic field, identified with the inverse non-commutativity parameter. Next we investigate the 3d ,,4 model with two non-commutative coordinates and explore its phase diagram. Our results agree with a conjecture by Gubser and Sondhi in d = 4, who predicted that the ordered regime splits into a uniform phase and a phase dominated by stripe patterns. We further present results for the correlators and the dispersion relation. In non-commutative field theory the Lorentz invariance is explicitly broken, which leads to a deformation of the dispersion relation. In one loop perturbation theory this deformation involves an additional infrared divergent term. Our data agree with this perturbative result. We also confirm the recent observation by Ambjø rn and Catterall that stripes occur even in d = 2, although they imply the spontaneous breaking of the translation symmetry. [source]

Continuum limits for classical sequential growth models

RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2010
Graham Brightwell
Abstract A random graph order, also known as a transitive percolation process, is defined by taking a random graph on the vertex set {0,,,n , 1} and putting i below j if there is a path i = i1,ik = j in the graph with i1 < , < ik. Rideout and Sorkin Phys. Rev. D 63 (2001) 104011 provide computational evidence that suitably normalized sequences of random graph orders have a "continuum limit." We confirm that this is the case and show that the continuum limit is always a semiorder. Transitive percolation processes are a special case of a more general class called classical sequential growth models. We give a number of results describing the large-scale structure of a general classical sequential growth model. We show that for any sufficiently large n, and any classical sequential growth model, there is a semiorder S on {0,,,n - 1} such that the random partial order on {0,,,n - 1} generated according to the model differs from S on an arbitrarily small proportion of pairs. We also show that, if any sequence of classical sequential growth models has a continuum limit, then this limit is (essentially) a semiorder. We give some examples of continuum limits that can occur. Classical sequential growth models were introduced as the only models satisfying certain properties making them suitable as discrete models for spacetime. Our results indicate that this class of models does not contain any that are good approximations to Minkowski space in any dimension , 2. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010 [source]

On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2005
Liliana Borcea
We consider finite difference approximations of solutions of inverse Sturm-Liouville problems in bounded intervals. Using three-point finite difference schemes, we discretize the equations on so-called optimal grids constructed as follows: For a staggered grid with 2 k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the Sturm-Liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known. A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal one, which is unknown. Fortunately, as we show here, the grid dependence on the unknown coefficients is weak, so the inversion can be done on a precomputed grid for an a priori guess of the unknown coefficients. This observation leads to a simple yet efficient inversion algorithm, which gives coefficients that converge pointwise to the true solution as the number k of data points tends to infinity. The cornerstone of our convergence proof is showing that optimal grids provide an implicit, natural regularization of the inverse problem, by giving reconstructions with uniformly bounded total variation. The analysis is based on a novel, explicit perturbation analysis of Lanczos recursions and on a discrete Gel'fand-Levitan formulation. © 2005 Wiley Periodicals, Inc. [source]

Continuum limits for classical sequential growth models

RANDOM STRUCTURES AND ALGORITHMS, Issue 2 2010
Graham Brightwell
Abstract A random graph order, also known as a transitive percolation process, is defined by taking a random graph on the vertex set {0,,,n , 1} and putting i below j if there is a path i = i1,ik = j in the graph with i1 < , < ik. Rideout and Sorkin Phys. Rev. D 63 (2001) 104011 provide computational evidence that suitably normalized sequences of random graph orders have a "continuum limit." We confirm that this is the case and show that the continuum limit is always a semiorder. Transitive percolation processes are a special case of a more general class called classical sequential growth models. We give a number of results describing the large-scale structure of a general classical sequential growth model. We show that for any sufficiently large n, and any classical sequential growth model, there is a semiorder S on {0,,,n - 1} such that the random partial order on {0,,,n - 1} generated according to the model differs from S on an arbitrarily small proportion of pairs. We also show that, if any sequence of classical sequential growth models has a continuum limit, then this limit is (essentially) a semiorder. We give some examples of continuum limits that can occur. Classical sequential growth models were introduced as the only models satisfying certain properties making them suitable as discrete models for spacetime. Our results indicate that this class of models does not contain any that are good approximations to Minkowski space in any dimension , 2. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010 [source]