Statistical Mechanics of Self-Avoiding Manifolds (Part II)

We consider a model of a D-dimensional tethered manifold interacting by excluded volume in R^d with a single point. Use of intrinsic distance geometry provides a rigorous definition of the analytic continuation of the perturbative expansion for arbitrary D, 0 < D < 2. Its one-loop renormalizability is first established by direct resummation. A renormalization operation R is then described, which ensures renormalizability to all orders. The similar question of the renormalizability of the self-avoiding manifold (SAM) Edwards model is then considered, first at one-loop, then to all orders. We describe a short-distance multi-local operator product expansion, which extends methods of local field theories to a large class of models with non-local singular interactions. It vindicates the direct renormalization method used earlier in part I of these lectures, as well as the corresponding scaling laws.Comment: 32 pages, 9 figures, Second Part and extensive update of Lecture Notes originally given in ``Statistical Mechanics of Membranes and Surfaces'', Fifth Jerusalem Winter School for Theoretical Physics (1987), D. R. Nelson, T. Piran,and S. Weinberg ed

Collapse transition of self-avoiding trails on the square lattice

The collapse transition of an isolated polymer has been modelled by many different approaches, including lattice models based on self-avoiding walks and self-avoiding trails. In two dimensions, previous simulations of kinetic growth trails, which map to a particular temperature of interacting self-avoiding trails, showed markedly different behaviour for what was argued to be the collapse transition than that which has been verified for models based of self-avoiding walks. On the other hand, it has been argued that kinetic growth trails represent a special simulation that does not give the correct picture of the standard equilibrium model. In this work we simulate the standard equilibrium interacting self-avoiding trail model on the square lattice up to lengths over $2,000,000$ steps and show that the results of the kinetic growth simulations are, in fact, entirely in accord with standard simulations of the temperature dependent model. In this way we verify that the collapse transition of interacting self-avoiding walks and trails are indeed in different universality classes in two dimensions

Geometry of the Casimir Effect

When the vacuum is partitioned by material boundaries with arbitrary shape, one can define the zero-point energy and the free energy of the electromagnetic waves in it: this can be done, independently of the nature of the boundaries, in the limit that they become perfect conductors, provided their curvature is finite. The first examples we consider are Casimir's original configuration of parallel plates, and the experimental situation of a sphere in front of a plate. For arbitrary geometries, we give an explicit expression for the zero-point energy and the free energy in terms of an integral kernel acting on the boundaries; it can be expanded in a convergent series interpreted as a succession of an even number of scatterings of a wave. The quantum and thermal fluctuations of vacuum then appear as a purely geometric property. The Casimir effect thus defined exists only owing to the electromagnetic nature of the field. It does not exist for thin foils with sharp folds, but Casimir forces between solid wedges are finite. We work out various applications: low temperature, high temperature where wrinkling constraints appear, stability of a plane foil, transfer of energy from one side of a curved boundary to the other, forces between distant conductors, special shapes (parallel plates, sphere, cylinder, honeycomb).Comment: 44 pages, 8 figures; Proceedings of the 15 th SIGRAV Conference on General Relativity and Gravitational Physics, Villa Mondragone, Monte Porzio Catone, Roma, Italy, September 9-12, 200

Continuously Varying Exponents for Oriented Self-Avoiding Walks

A two-dimensional conformal field theory with a conserved $U(1)$ current $\vec J$, when perturbed by the operator ${\vec J}^{\,2}$, exhibits a line of fixed points along which the scaling dimensions of the operators with non-zero $U(1)$ charge vary continuously. This result is applied to the problem of oriented polymers (self-avoiding walks) in which the short-range repulsive interactions between two segments depend on their relative orientation. While the exponent $\nu$ describing the fractal dimension of such walks remains fixed, the exponent $\gamma$, which gives the total number $\sim N^{\gamma-1}\mu^N$ of such walks, is predicted to vary continuously with the energy difference.Comment: 15 pages, plain TeX, 2 uuencoded postscript figures, OUTP-93-41

Renormalization Group Approach to Interacting Crumpled Surfaces: The hierarchical recursion

We study the scaling limit of a model of a tethered crumpled D-dimensional random surface interacting through an exclusion condition with a fixed impurity in d-dimensional Euclidean space by the methods of Wilson's renormalization group. In this paper we consider a hierarchical version of the model and we prove rigorously the existence of the scaling limit and convergence to a non-Gaussian fixed point for $1 \leq D0$ sufficiently small, where $\epsilon = D - (2-D) {d\over 2}$.Comment: 47 pages in simple Latex, PAR-LPTHE 934

Liouville Quantum Gravity and KPZ

Consider a bounded planar domain D, an instance h of the Gaussian free field on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma < 2. The Liouville quantum gravity measure on D is the weak limit as epsilon tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h on the circle of radius epsilon centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the KPZ relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of the boundary of D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.Comment: 56 pages. Revised version contains more details. To appear in Inventione

Operator Product Expansion on a Fractal: The Short Chain Expansion for Polymer Networks

We prove to all orders of renormalized perturbative polymer field theory the existence of a short chain expansion applying to polymer solutions of long and short chains. For a general polymer network with long and short chains we show factorization of its partition sum by a short chain factor and a long chain factor in the short chain limit. This corresponds to an expansion for short distance along the fractal perimeter of the polymer chains connecting the vertices and is related to a large mass expansion of field theory. The scaling of the second virial coefficient for bimodal solutions is explained. Our method also applies to the correlations of the multifractal measure of harmonic diffusion onto an absorbing polymer. We give a result for expanding these correlations for short distance along the fractal carrier of the measure.Comment: 28 pages, revtex, 4 Postscript figures, 3 latex emlines pictures. Replacement eliminates conflict with a blob resul

Two-Dimensional Copolymers and Exact Conformal Multifractality

We consider in two dimensions the most general star-shaped copolymer, mixing random (RW) or self-avoiding walks (SAW) with specific interactions thereof. Its exact bulk or boundary conformal scaling dimensions in the plane are all derived from an algebraic structure existing on a random lattice (2D quantum gravity). The multifractal dimensions of the harmonic measure of a 2D RW or SAW are conformal dimensions of certain star copolymers, here calculated exactly as non rational algebraic numbers. The associated multifractal function f(alpha) are found to be identical for a random walk or a SAW in 2D. These are the first examples of exact conformal multifractality in two dimensions.Comment: 4 pages, 2 figures, revtex, to appear in Phys. Rev. Lett., January 199

Boundary Correlators in 2D Quantum Gravity: Liouville versus Discrete Approach

We calculate a class of two-point boundary correlators in 2D quantum gravity using its microscopic realization as loop gas on a random surface. We find a perfect agreement with the two-point boundary correlation function in Liouville theory, obtained by V. Fateev, A. Zamolodchikov and Al. Zamolodchikov. We also give a geometrical meaning of the functional equation satisfied by this two-point function.Comment: 21 pages, 5 figures, harvmac, eqs. (2.11) and (5.11) correcte