The number and size of branched polymers in high dimensions

We consider two models of branched polymers (lattice trees) on thed-dimensional hypercubic lattice: (i)the nearest-neighbor model in sufficiently high dimensions, and (ii) a spread-out or long-range model ford>8, in which trees are constructed from bonds of length less than or equal to a large parameterL. We prove that for either model the critical exponent for the number of branched polymers exists and equals 5/2, and that the critical exponentv for the radius of gyration exists and equals 1/4. This improves our earlier results for the corresponding generating functions. The proof uses the lace expansion, together with an analysis involving fractional derivatives which has been applied previously to the self-avoiding walk in a similar context.     

Multifractal analysis of Brownian zero set

We present a method for the derivation of the generating function and computation of critical exponents for several cluster models (staircase, bar-graph, and directed column-convex polygons, as well as partially directed self-avoiding walks), starting with nonlinear functional equations for the generating function. By linearizing these equations, we first give a derivation of the generating functions. The nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz. Alternatively, taking the continuum limit leads to nonlinear differential equations, from which one can extract the scaling function. We find that all the above models are in the same universality class with exponents _u =-1/2, _i =-1/3, and =2/3. All models have as their scaling function the logarithmic derivative of the Airy function.     

Biased Diffusion in a One-Dimensional Adsorbed Monolayer

We study the dynamics of a probe particle, which performs biased diffusive motion in a one-dimensional adsorbed monolayer composed of mobile hard-core particles undergoing continuous exchanges with a vapor phase. In terms of a mean-field-type approach, based on the decoupling of the third-order correlation functions into the product of pairwise correlations, we determine analytically the density profiles of the monolayer particles, as seen from the stationary moving probe, and calculate the terminal velocity V _pr, mobility _pr and the self-diffusion coefficient D _pr of the probe. Our analytical results are confirmed by Monte Carlo simulations.     

Higher Conformal Multifractality

We derive, from conformal invariance and quantum gravity, the multifractal spectrum f() of the harmonic measure (i.e., electrostatic potential, or diffusion field) near any conformally invariant fractal in two dimensions. It gives the Hausdorff dimension of the set of points where the potential varies with distance r to the fractal frontier as r . First examples are a random walk, i.e., a Brownian motion, a self-avoiding walk, or a critical percolation cluster. The generalized dimensions D(n) as well as the multifractal functions f() are derived, and are all identical for these three cases. The external frontiers of a Brownian motion and of a percolation cluster are thus identical to a self-avoiding walk in the scaling limit. The multifractal (MF) function f(,c) of the electrostatic potential near any conformally invariant fractal boundary, like a critical O(N) loop or a Q-state Potts cluster, is given as a function of the central charge c of the associated conformal field theory. The dimensions D _EP of the external perimeter and D _H of the hull of a critical scaling curve or cluster obey the superuniversal duality equation . Finally, for a conformally invariant scaling curve which is simple, i.e., without double points, we derive higher multifractal functions, like the universal function f ₂(,) which gives the Hausdorff dimension of the points where the potential varies jointly with distance r as r on one side of the curve, and as r on the other. The general case of the potential distribution between the branches of a star made of an arbitrary number of scaling paths is also treated. The results apply to critical O(N) loops, Potts clusters, and to the SLE process. We present a duality between external perimeters of Potts clusters and O(N) loops at their critical point, as well as the corresponding duality in the SLE process for =16.     

Branched Polymers on the Two-Dimensional Square Lattice with Attractive Surfaces

Using the renormalization group approach, an analysis is given of the asymptotic properties of branched polymers situated on the two-dimensional square lattice with attractive impenetrable surfaces. We modeled branched polymers as site lattice animals with loops and site lattice animals without loops on the simple square lattice. We found the gyration radius critical exponent =0.6511±0.0003 and =0.6513±0.0003 for branched polymers with and without loops, respectively. Our results for the crossover exponent =0.502±0.003 for branched polymers with loops and =0.503±0.003 for branched polymers without loops satisfy the recent hyperuniversality conjecture = . In addition, we have studied partially directed site lattice animals.     

The long-time behavior of initially separated A+B→0 reaction-diffusion systems with arbitrary diffusion constants

We examine the long-time behavior of A+B0 reaction-diffusion systems with initially segregated species A and B. All of our analysis is carried out for arbitrary (positive) values of the diffusion constantsD _A andD _B and initial concentrationsa ₀ andb ₀ of A's and B's. We divide the domain of the partial differential equations describing the problem into several regions in which they can be reduced to simpler, solvable equations, and we merge the solutions. Thus we derive general formulas for the concentration profiles outside the reaction zone, the location of the reaction zone center, and the total reaction rate. An asymptotic condition for the reaction front to be stationary is also derived. The properties of the reaction layer are studied in the mean-field approximation, and we show that not only the scaling exponents, but also the scaling functions are independent ofD _A,D _B,a ₀ andb ₀.     

Dynamics of selfavoiding tethered membranes. II. Inclusion of hydrodynamic interaction (Zimm model)

The dynamical scaling properties of selfavoiding polymerized membranes with internal dimension D embedded into d dimensions are studied including hydrodynamical interactions. It is shown that the theory is renormalizable to all orders in perturbation theory and that the dynamical scaling exponent z is given by z=d. The crossover to the region, where the membrane is crumpled swollen but the hydrodynamic interaction irrelevant is discussed. The results apply as well to polymers (D=1) as to membranes (D=2). Received: 5 September 1997 / Accepted: 17 November 1997     

Reduction Formula for Fermion Loops and Density Correlations of the 1D Fermi Gas

Fermion N-loops with an arbitrary number of density vertices N>d+1 in d spatial dimensions can be expressed as a linear combination of (d+1)-loops with coefficients that are rational functions of external momentum and energy variables. A theorem on symmetrized products then implies that divergences of single loops for low energy and small momenta cancel each other when loops with permuted external variables are summed. We apply these results to the one-dimensional Fermi gas, where an explicit formula for arbitrary N-loops can be derived. The symmetrized N-loop, which describes the dynamical N-point density correlations of the 1D Fermi gas, does not diverge for low energies and small momenta. We derive the precise scaling behavior of the symmetrized N-loop in various important infrared limits.     

Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A ^–1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that . We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.     

Correlation functions of conserved currents in four dimensional conformal field theory

We derive a generating function for all the 3-point functions of higher spin conserved currents in four dimensional conformal field theory. The resulting expressions have a rather surprising factorized form which suggests that they can all be realized by currents built from free massless fields of arbitrary (half-)integer spin s  . This property is however not necessarily true also for the higher-point functions. As an illustration we analyze the general 4-point function of conserved abelian U(1)$U(1)$ currents of scale dimension equal to three and find that apart from the two free field realizations there is a unique possible function which may correspond to an interacting theory. Although this function passes several non-trivial consistency tests, it remains an open challenging problem whether it can be actually realized in an interacting CFT.     

Statistical mechanics and stability of random lattice field theory

The averaging procedure in the random lattice field theory is studied by viewing it as a statistical mechanics of a system of classical particles. The corresponding thermodynamic phase is shown to determine the random lattice configuration which contributes dominantly to the generating function. The non-abelian gauge theory in four (space plus time) dimensions in the annealed and quenched averaging versions is shown to exist as an ideal classical gas, implying that macroscopically homogeneous configurations dominate the configurational averaging. For the free massless scalar field theory with O(n) global symmetry, in the annealed average, the pressure becomes negative for dimensions greater than two when n exceeds a critical number. This implies that macroscopically inhomogeneous collapsed configurations contribute dominantly. In the quenched averaging, the collapse of the massless scalar field theory is prevented and the system becomes an ideal gas which is at infinite temperature. Our results are obtained using exact scaling analysis. We also show approximately that SU(N) gauge theory collapses for dimensions greater than four in the annealed average. Within the same approximation, the collapse is prevented in the quenched average. We also obtain exact scaling differential equations satisfied by the generating function and physical quantities.     

A renormalization group analysis of turbulent transport

The field-theoretic renormalization group is used to derive scaling relations for the transport of passive scalars by an incompressible velocity field with a specified energy spectrum. Results are obtained with the analog of the expansion of critical phenomena and compared to exact results which are available for shear flows in two dimensions.A 1/N expansion is proposed for the regions in which the expansion fails.     

Variational evaluation of correlation functions for lattice electrons in high dimensions

We present a general formalism for the diagrammatic calculation of correlation functions for Hubbard-type models in terms of projected wave functions. It is shown that in the limit of high spatial dimensionsd only diagrams with bubble-structure remain. This causes correlation functions to have an overall RPA-type form ind. Exact evaluations are performed for the Gutzwiller wave function. Nearest neighbor correlations are shown to be proportional to their value in the non-interacting case, i.e. are renormalized. However, their absolute value is only of order 1/d. Hence this wave function does not describe spin correlations adequately in high dimensions. The asymptotic behavior of the spin-correlation function is extracted and is found to have a scaling form similar tod=1. Assuming this form to hold in all dimensions we show that the Brinkman-Rice transition only occurs ind=. Finite orders of perturbation theory in 1/d around this singular point are not sufficient to remove the transition.     

Microcanonical density functionals for critical systems: An exact one-dimensional example

Free-energy functionals suitable for describing realistic, nonuniform systems near criticality are discussed with emphasis on the advantages of a local formalism. It is proposed to investigatemicro canonical functionals in which both the usual order-parameter (or magnetization) density m(r)and the local energy density (r), which has independent critical fluctuations, are employed. This approach is tested by an exact calculation of the microcanonical functional [{m}, {}] in the continuum limit for a one-dimensional Ising model. Remarkably, the microcanonical functional is found to be local irrespective of the proximity to the critical point (located at zero temperature and zero field). Furthermore, its form relates closely to the scaling postulate advanced earlier by de Gennes and Fisher and displays features of conformal covariance.     

On the critical properties of the Edwards and the self-avoiding walk model of polymer chains

We study the two- and three-dimensional, superrenormalizable Edwards model and the self-avoiding walk model of polymers. Using a Schwinger-Dyson equation and upper and lower bounds on correlations in terms of “skeleton diagrams” [6] we establish the existence of a non-trivial continuum limit in the two- and three-dimensional, superrenormalizable Edwards model. We also prove that perturbation theory is asymptotic for the continuum correlations of these models.A fairly detailed analysis of the approach to the critical point in the self-avoiding walk model is presented. In particular, we show that η<1. In dimension d?4, we discuss rigorous consequences of the conjecture that η is non-negative: among other implications, we derive that the continuum limit is trivial and that γ=1, in d?5 dimensions, and that corrections to mean-field scaling laws are at most logarithmic in four dimensions.     

The Bloch-Okounkov Correlation Functions of Classical Type

Bloch and Okounkov introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. This function has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, etc., and it can also be interpreted as correlation functions on integrable -modules of level one. Such -correlation functions at higher levels were then calculated by Cheng and Wang. In this paper, generalizing the type A results, we formulate and determine the n-point correlation functions in the sense of Bloch-Okounkov on integrable modules over classical Lie subalgebras of of type B, C, D at arbitrary levels. As byproducts, we obtain new q-dimension formulas for integrable modules of type B, C, D and some fermionic type q-identities.     

Fractal dimensions of time sequences

We present a simple and efficient way for calculating the fractal dimension $D$ of any time sequence sampled at a constant time interval. We calculated the error of a piecewise interpolation to $N+1$ points of the time sequence with respect to the next level of ($2N+1$)-point interpolation. This error was found to be proportional to the scale (i.e., $1/N$) to the power of $1?D$. A simple analysis showed that our method is equivalent to the inverse process of the method of random midpoint displacement widely used in generating fractal Brownian motion for a given $D$. The efficiency of our method makes the fractal dimension a practical tool in analyzing the abundant data in natural, economic, and social sciences.     

Infrared behaviour of metallic systems in one,two and three dimensions

A simple approximate expression for the electron lifetime() in metals is rederived and discussed for different dimensions. In the 3D-case we get the well known Drude behaviour, i.e. a constant. In one dimension() is strongly frequency-dependent in the IR. The 2D-case is intermediate to the preceding ones. These results are essentially due to the different form of the Fermi surface for an electron gas in one, two and three dimensions.     

Infrared behaviour of metallic systems in one, two and three dimensions

A simple approximate expression for the electron lifetime() in metals is rederived and discussed for different dimensions. In the 3D-case we get the well known Drude behaviour, i.e. a constant. In one dimension() is strongly frequency-dependent in the IR. The 2D-case is intermediate to the preceding ones. These results are essentially due to the different form of the Fermi surface for an electron gas in one, two and three dimensions.