289 research outputs found
Liouville quantum gravity with matter central charge in : a probabilistic approach
There is a substantial literature concerning Liouville quantum gravity (LQG)
in two dimensions with conformal matter field of central charge
. Via the DDK ansatz, LQG can
equivalently be described as the random geometry obtained by exponentiating
times a variant of the planar Gaussian free field (GFF), where
satisfies .
Physics considerations suggest that LQG should also make sense in the regime
when . However, the behavior in this regime is rather
mysterious in part because the corresponding value of is complex, so
analytic continuations of various formulas give complex answers which are
difficult to interpret in a probabilistic setting.
We introduce and study a discretization of LQG which makes sense for all
values of . Our discretization consists
of a random planar map, defined as the adjacency graph of a tiling of the plane
by dyadic squares which all have approximately the same "LQG size" with respect
to the GFF. We prove that several formulas for dimension-related quantities are
still valid for , with the caveat that the
dimension is infinite when the formulas give a complex answer. In particular,
we prove an extension of the (geometric) KPZ formula for , which gives a finite quantum dimension iff the Euclidean
dimension is at most . We also show that the
graph distance between typical points with respect to our discrete model grows
polynomially whereas the cardinality of a graph distance ball of radius
grows faster than any power of (which suggests that the Hausdorff dimension
of LQG is infinite for ).
We include a substantial list of open problems.Comment: 53 pages, 6 figure
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Efficient Computer Simulation of Polymer Conformation. I. Geometric Properties of the Hard-Sphere Model
A system of efficient computer programs has been developed for simulating the conformations of macromolecules. The conformation of an individual polymer is defined as a point in conformation space, whose mutually orthogonal axes represent the successive dihedral angles of the backbone chain. The statistical-mechanical average of any property is obtained as the usual configuration integral over this space. A Monte Carlo method for estimating averages is used because of the impossibility of direct numerical integration. Monte Carlo corresponds to the execution of a Markoffian random walk of a representative point through the conformation space. Unlike many previous Monte Carlo studies of polymers, which sample conformation space indiscriminately, importance sampling increases efficiency because selection of new polymers is biased to reflect their Boltzmann probabilities in the canonical ensemble, leading to reduction of sampling variance and hence to greater accuracy! in given computing time. The simulation is illustrated in detail. Overall running time is proportional to n^(5/4), where n is the chain length. Results are presented for a hard-sphere linear polymer of n atoms, with free dihedral rotation, with n = 20-298. The fraction of polymers accepted in the importance sampling scheme, fA, is fit to a Fisher-Sykes attrition relation, giving an effective attrition constant of zero. fA is itself an upper bound to the partition function, Q, relative to the unrestricted walk. The mean-squared end-to-end distance and radius of gyration exhibit the expected exponential dependence, but with exponent for the radius of gyration significantly greater than that of the end-to-end distance. The 90% confidence limits calculated for both exponents did not include either 6/5 or 4/3, the lattice and zero-order perturbation values, respectively. A self-correcting scheme for generating coordinates free of roundoff error is given in an Appendix
Computational methods in string and field theory
Thesis is submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the University of the Witwatersrand, Faculty of Science, School of Physics, University of the Witwatersrand, Johannesburg, 2018Like any field or topic of research, significant advancements can be made with increasing computational
power - string theory is no exception. In this thesis, an analysis is performed within three areas: Calabi–Yau
manifolds, cosmological inflation and application of conformal field theory. Critical superstring theory is a ten
dimensional theory. Four of the dimensions refer to the spacetime dimensions we see in nature. To account
for the remaining six, Calabi-Yau manifolds are used. Knowing how the space of Calabi-Yau manifolds
is distributed gives valuable insight into the compactification process. Using computational modeling and
statistical analysis, previously unseen patterns of the distribution of the Hodge numbers are found. In
particular, patterns in frequencies exhibit striking new patterns - pseudo-Voigt and Planckian distributions
with high confidence and exact fits for many substructures. The patterns indicate typicality within the
landscape of Calabi–Yau manifolds of various dimensions. Inflation describes the exponential expansion of
the universe after the Big Bang. Finding a successful theory of inflation centres around building a potential
of the inflationary field, such that it satisfies the slow-roll conditions. The numerous ways this can be done,
coupled with the fact that each model is highly sensitive to initial conditions, means an analytic approach
is often not feasible. To bypass this, a statistical analysis of a landscape of thousands of random single and
multifield polynomial potentials is performed. Investigation of the single field case illustrates a window in
which the potentials satisfy the slow-roll conditions. When there are two scalar fields, it is found that the
probability depends on the choice of distribution for the coefficients. A uniform distribution yields a 0.05%
probability of finding a suitable minimum in the random potential whereas a maximum entropy distribution
yields a 0.1% probability. The benefit of developing computational tools extends into the interdisciplinary
study between conformal field theory and the theory of how wildfires propagate. Using the two dimensional
Ising model as a basis of inspiration, computational methods of analyzing how fires propagate provide a new
tool set which aids in the process of both modeling large scale wildfires as well as describing the emergent
scale invariant structure of these fires. By computing the two point and three point correlations of fire
occurrences in particular regions within Botswana and Kazakhstan, it is shown that this proposed model
gives excellent fits, with the model amplitude being directly proportional to the total burn area of a particular
year.EM201
Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
We reconsider the conceptual foundations of the renormalization-group (RG)
formalism, and prove some rigorous theorems on the regularity properties and
possible pathologies of the RG map. Regarding regularity, we show that the RG
map, defined on a suitable space of interactions (= formal Hamiltonians), is
always single-valued and Lipschitz continuous on its domain of definition. This
rules out a recently proposed scenario for the RG description of first-order
phase transitions. On the pathological side, we make rigorous some arguments of
Griffiths, Pearce and Israel, and prove in several cases that the renormalized
measure is not a Gibbs measure for any reasonable interaction. This means that
the RG map is ill-defined, and that the conventional RG description of
first-order phase transitions is not universally valid. For decimation or
Kadanoff transformations applied to the Ising model in dimension ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . We discuss in detail
the distinction between Gibbsian and non-Gibbsian measures, and give a rather
complete catalogue of the known examples. Finally, we discuss the heuristic and
numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also
ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.
Non-local meta-conformal invariance, diffusion-limited erosion and the XXZ chain
Diffusion-limited erosion is a distinct universality class of fluctuating
interfaces. Although its dynamical exponent , none of the known variants
of conformal invariance can act as its dynamical symmetry. In spatial
dimensions, its infinite-dimensional dynamic symmetry is constructed and shown
to be isomorphic to the direct sum of three loop-Virasoro algebras, with the
maximal finite-dimensional sub-algebra
.
The infinitesimal generators are spatially non-local and use the Riesz-Feller
fractional derivative. Co-variant two-time response functions are derived and
reproduce the exact solution of diffusion-limited erosion. The relationship
with the terrace-step-kind model of vicinal surfaces and the integrable XXZ
chain are discussed.Comment: Latex 2e, 28 pp, 4 figures (revised, with 2 new figures
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