Random Manifolds Research Articles

Stability of the Mézard-Parisi Solution for Random Manifolds

The eigenvalues of the Hessian associated with random manifolds are constructed for the general case of $R$ steps of replica symmetry breaking. For the Parisi limit $R\to\infty$ (continuum replica symmetry breaking) which is relevant for the manifold dimension $D<2$, they are shown to be non negative.

Elastic chain in a random potential: Simulation of the displacement correlation function and relaxation.

We simulate the low temperature behaviour of an elastic chain in a random potential where the displacements $u(x)$ are confined to the {\it longitudinal} direction ($u(x)$ parallel to $x$) as in a one dimensional charge density wave--type problem. We calculate the displacement correlation function $g(x)=< (u(x)-u(0))^2>$ and the size dependent average square displacement $W(L)=<(u(x)-\bar{u})^2>$. We find that $g(x)\sim x^{2\eta}$ with $\eta\simeq3/4$ at short distances and $\eta\simeq3/5$ at intermediate distances. We cannot resolve the asymptotic long distance dependence of $g$ upon $x$. For the system sizes considered we find $g(L/2)\propto W\sim L^{2\chi}$ with $\chi\simeq2/3$. The exponent $\eta\simeq3/5$ is in agreement with the Random Manifold exponent obtained from replica calculations and the exponent $\chi\simeq2/3$ is consistent with an exact solution for the chain with {\it transverse} displacements ($u(x)$ perpendicular to $x$).The distribution of nearest distances between pinning wells and chain-particles is found to develop forbidden regions.

Banach-Tarski theorem and Cantorian micro space-time

Starting from the cosmic initial singularity scenario and general relativity, Banach-Tarski theorem together with the theorem of Mauldin-Williams leads to the conclusion that micro space should resemble a realization of a random quasi-transfinite four-dimensional Cantorian manifold ( d c (4))with a Golden Mean Hausdorff dimension at the core ( d c (0)) where d c (0) = ¦〈d c (0)¦d c (0)〉¦ and 〈d c (0)¦ = 0 − id c (0) . In the imaginary phase space the dimensionality (the number of components) is eight and is related to SU(3) and quarks.

Elastic theory of flux lattices in the presence of weak disorder.

The effect of disorder on flux lattices at equilibrium is studied quantitatively in the absence of free dislocations using both the Gaussian variational method and the renormalization group. Our results for the mean square relative displacements clarify the nature of the crossovers with distance. We find three regimes: (i) a short distance regime (``Larkin regime'') where elasticity holds (ii) an intermediate regime (``Random Manifold'') where vortices are pinned independently (iii) a large distance, quasi-ordered regime where the periodicity of the lattice becomes important and there is universal logarithmic growth of displacements for $2<d<4$ and persistence of algebraic quasi-long range translational order. The functional renormalization group to $O(\epsilon=4-d)$ and the variational method, agree within $10\%$ on the value of the exponent. In $d=3$ we compute the crossover function between the three regimes. We discuss the observable signature of this crossover in decoration experiments and in neutron diffraction experiments on flux lattices. Qualitative arguments are given suggesting the existence for weak disorder in $d=3$ of a `` Bragg glass '' phase without free dislocations and with algebraically divergent Bragg peaks. In $d=1+1$ both the variational method and the Cardy-Ostlund renormalization group predict a glassy state below the same transition temperature $T=T_c$, but with different behaviors. Applications to $d=2+0$ systems and experiments on magnetic bubbles are discussed.

Random geometries and real space renormalization group

A method of ``blocking'' triangulations that rests on the self-similarity feature of dynamically triangulated random manifolds is proposed. The method is used to define the renormalization group for random geometries. As an illustration, the idea is applied to pure euclidean quantum gravity in two dimensions. Generalisation to more complicated systems and to higher dimensionalities of space-time appears straightforward.

Stochastic inertial manifold

A nonlinear stochastic evolution equation in Hilbert space with generalized additive white noise is considered. A concept of stochastic mertial manifold is introduced, defined as a random manifold depending on time, which is finite dimensional, invariant for the dynamic, and attracts exponentially fast all the trajectories as t → ∞. Under the classical spectral gap condition of the deterministic theory, the existence of a stochastic inertial manifold is proved. It is obtained as the solution of a stochastic partial differential equation of degenerate parabolic type, studied by a variant of Bernstein method. A result of existence and uniqueness of a stationary inertial manifold is also proved; the stationary inertial manifold contains the random attractor, introduced in previous works.

Brushes formed by self‐similarly branched polymers and random manifolds

AbstractWhen randomly branched polymers are grafted to a surface, polymer brushes are the result, similarly as well known in the case of linear grafted polymer chains. These brushes behave, however, special, as severe restrictions in the conformation of individual polymers are present. These restrictions are firstly due to the high degree of branching and secondly to the natural maximum stretching ratio of the branched molecules, that is given by Hmax = aM1/D. a is the typical size of a monomeric unit, D is the spectral dimension and M the total mass of the branched polymer. Brushes cannot exceed this height, and limits on the grafting density as functions of the spectral dimension are investigated. For all thermodynamic situations the minimum area per chain scales as σmin = aM(D − 1)/D.

Pinned vortex lattices: a variational study

We study the effects of disorder on a flux lattice at equilibrium, in the absence of free dislocations, using a gaussian variational ansatz. This variational method allows us to obtain the physical properties of such a system in any dimension. For the case of point like disorder, we find universal logarithmic growth of displacements for 2 < d < 4: 〈u(χ) − u(0)〉 2 ∼ A d log|χ| and persistence of algebraic quasi-long range translational order. Close to four dimensions, a functional renormalization group calculation in ϵ = 4 − d agrees within 10% on the value of A d with the variational method. We compute the function describing the crossover between the short distance (“random manifold”) regime and the logarithmic regime. In d = 2, the variational method gives a logarithmic growth of displacement with a temperature independent prefactor at variance from previous RG calculations. Columnar disorder is studied by the same method. In d=2+1, describing vortex lattices with the field along the columns, one finds a logarithmic growth of the in-plane displacements. In d = 1 + 1 dimensions, this problem is related by bosonization to Anderson localization of interacting electrons by static disorder. The variational method predicts a localized-delocalized transition, for attractive enough interactions, in agreement with previous studies, as well as a conductivity σ( ω) ∼ ω 2 in the localized phase.

Statistical physics of fault patterns self-organized by repeated earthquakes

This work presents at attempt to model brittle ruptures and slips in a continental plate and its spontaneous organization by repeated earthquakes in terms of coarse-grained properties of the mechanical plate. A statistical physics model, which simulates anti-plane shear deformation of a thin plate with inhomogeneous elastic properties, is thus analyzed theoretically and numerically in order to study the spatio-temporal evolution of rupture patterns in response to a constant applied strain rate at its borders, mimicking the effect of neighboring plates. Rupture occurs when the local stress reaches a threshold value. Broken elements are instantaneously healed and retain the original material properties, enabling the occurrence of recurrent earthquakes. Extending previous works (Cowieet al., 1993;Miltenbergeret al., 1993), we present a study of the most startling feature of this model which is that ruptures become strongly correlated in space and time leading to the spontaneous development of multifractal structures and gradually accumulate large displacements. The formation of the structures and the temporal variation of rupture activity is due to a complex interplay between the random structure, long-range elastic interactions and the threshold nature of rupture physics. The spontaneous formation of fractal fault structures by repeated earthquakes is mirrored at short times by the spatio-temporal chaotic dynamics of earthquakes, well-described by a Gutenberg-Richter power law. We also show that the fault structures can be understood as pure geometrical objects, namely minimal manifolds, which in two dimensions correspond to the random directed polymer (RDP) problem. This mapping allows us to use the results of many studies on the RDP in the field of statistical physics, where it is an exact result that the minimal random manifolds in 2D systems are self-affine with a roughness exponent 2/3. We also present results pertaining to the influence of the degree β of stress release per earthquake on the competition between faults. Our results provide a rigorous framework from which to initiate rationalization of many, reported fractal fault studies.

Elastic theory of pinned flux lattices.

The effect of weak impurity disorder on flux lattices at equilibrium is studied in the absence of free dislocations using both the Gaussian variational method and, to O(\ensuremath{\epsilon}=4-d), the functional renormalization group. We find universal logarithmic growth of displacements for 2d4:〈u(x)-u(0)${\mathrm{〉}}^{2}$\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\sim}${\mathit{A}}_{\mathit{d}}$ln\ensuremath{\Vert}x\ensuremath{\Vert} and persistence of algebraic quasi-long-range translational order. When the two methods can be compared they agree within 10% on the value of ${\mathit{A}}_{\mathit{d}}$. We compute the function describing the crossover between the ``random manifold'' regime and the logarithmic regime. A similar crossover could be observable in present decoration experiments.

Four-dimensional dynamically triangulated gravity coupled to matter.

We investigate the phase structure of four-dimensional quantum gravity coupled to Ising spins or Gaussian scalar fields by means of numerical simulations. The quantum gravity part is modelled by the summation over random simplicial manifolds, and the matter fields are located in the center of the 4-simplices, which constitute the building blocks of the manifolds. We find that the coupling between spin and geometry is weak away from the critical point of the Ising model. At the critical point there is clear coupling, which qualitatively agrees with that of gaussian fields coupled to gravity. In the case of pure gravity a transition between a phase with highly connected geometry and a phase with very ``dilute'' geometry has been observed earlier. The nature of this transition seems unaltered when matter fields are included. It was the hope that continuum physics could be extracted at the transition between the two types of geometries. The coupling to matter fields, at least in the form discussed in this paper, seems not to improve the scaling of the curvature at the transition point.

Renormalization theory for interacting crumpled manifolds

We consider a continuous model of D-dimensional elastic (polymerized) manifold fluctuating in d-dimensional euclidean space, interacting with a single impurity via an attractive or repulsive δ-potential (but without self-avoidance interactions). Except for D = 1 (the polymer case), this model cannot be mapped onto a loca; field theory. We show that the use of intrinsic distance geometry allows for a rigorous construction of the high-temperature perturbative expansion and for analytic continuation in the manifold dimension D. We study the renormalization properties of the model for 0 < D < 2, and show that for bulk space dimension d smaller that the upper critical dimension d ★ = 2D (2−D) , the perturbative expansion is ultraviolet finite, while ultraviolet divergences occur as pole at d = d ★. The standard proof of perturbative renormalizability for local field theories (the Bogoliubov-Parasiuk-Hepp theorem) does not apply to this model. We prove perturbative renormalizability to all orders by constructing a subtraction operator R based on a generalization of the Zimmermann forests formalism, and which makes the theory finite at d = d ★. This subtraction operation corresponds to a renormalization of the coupling constant of the model (strength of the interaction with the impurity). The existence of a Wilson function, of an ϵ-expansion à la Wilson-Fisher around the critical dimension, of scaling laws for d < d ★ in the repulsive case, and of non-trivial critical exponents of the delocalization transition for d > d ★ in the attractive case, is thus established. To our knowledge, this study provides the first proof of renormalizability for a model of extended objects, and should be applicable to the study of self-avoidance interactions for random manifolds.

Quantum gravity, dynamical triangulations and higher-derivative regularization

We consider a discrete model of euclidean quantum gravity in four dimensions based on a summation over random simplicial manifolds. The action used is the Einstein-Hilbert action plus an R 2 term. The phase diagram as a function of the bare coupling constants is studied in the search for a sensible continuum limit. For small values of the coupling constant of the R 2 term the model seems to belong to the same universality class as the model with pure Einstein-Hilbert action and exhibits the same phase transition. The order of the transition may be second or higher. The average curvature is positive at the phase transition, which makes it difficult to understand the possible scaling relations of the model.

3D quantum gravity coupled to matter

We investigate the phase structure of three-dimensional quantum gravity coupled to an Ising spin system by means of numerical simulations. The quantum gravity part is modelled by the summation over random simplicial manifolds, and the Ising spins are located in the center of the tetrahedra, which constitute the building blocks of the piecewise linear manifolds. We find that the coupling between spin and geometry is weak away from the critical point of the Ising model. At the critical point there is clear coupling, which, however, does not seem to change the first order transition between the “hot” and “cold” phase of three dimensional simplicial quantum gravity observed earlier.

Three-dimensional simplicial quantum gravity

We consider a discrete model of euclidean quantum gravity in three dimensions based on a summation over random simplicial manifolds. The phase diagram as a function of the cosmological coupling constant and the gravitational coupling constant is studied in the search for a sensible continuum limit. An interesting cross-over behaviour is observed, where the universes changes from being crumpled, of essentially zero radius, to being extended objects.

Four-dimensional simplicial quantum gravity

We consider a discrete model of euclidean quantum gravity in four dimensions based on a summation over random simplicial manifolds. The phase diagram as a function of the bare gravitational coupling constant is studied in the search for a sensible continuum limit.

Mean field dynamics of random manifolds

Mean field dynamics of random manifolds

Replica field theory for random manifolds

We consider the field theory formulation for manifolds in random media using the replica method. We use a variational (Hartree-Fock like) method which shows that replica symmetry is spontaneously broken. A hierarchical breaking of symmetry allows one to take into account the existence of many metastable states for the manifold, and to recover the results of the Flory scaling arguments for the wandering exponent. This field theoretic derivation of Flory results opens the way to computing corrections to these exponents.

THREE-DIMENSIONAL SIMPLICIAL QUANTUM GRAVITY AND GENERALIZED MATRIX MODELS

We consider a discrete model of Euclidean quantum gravity in three dimensions based on a summation over random simplicial manifolds. We derive some elementary properties of the model and discuss possible “matrix” models for 3-D gravity.

Random manifolds in the high-temperature expansion of spin and gauge models

The author writes down explicit expressions for the high-temperature expansion of several spin and gauge models with logarithmic action, defined on certain lattices. Such lattices exist in any dimension for spin models, but in the gauge case their existence is only proved modulo a technical assumption. The partition function of the spin models is a gas of closed loops, weighted by NL, where N is the number of spin components and L is the total length of the loops. The gauge model is a gas of closed, self-avoiding surfaces, weighted by Nchi , where N is the dimension of the representation and chi is the Euler characteristic.