Branched Polymers on the Given-Mandelbrot family of fractals

We study the average number A_n per site of the number of different configurations of a branched polymer of n bonds on the Given-Mandelbrot family of fractals using exact real-space renormalization. Different members of the family are characterized by an integer parameter b, b > 1. The fractal dimension varies from $log_{_2} 3$ to 2 as b is varied from 2 to infinity. We find that for all b > 2, A_n varies as $\lambda^n exp(b n ^{\psi})$, where $\lambda$ and $b$ are some constants, and $0 < \psi <1$. We determine the exponent $\psi$, and the size exponent $\nu$ (average diameter of polymer varies as $n^\nu$), exactly for all b > 2. This generalizes the earlier results of Knezevic and Vannimenus for b = 3 [Phys. Rev {\bf B 35} (1987) 4988].Comment: 24 pages, 8 figure

Convex lattice polygons of fixed area with perimeter dependent weights

We study fully convex polygons with a given area, and variable perimeter length on square and hexagonal lattices. We attach a weight t^m to a convex polygon of perimeter m and show that the sum of weights of all polygons with a fixed area s varies as s^{-theta_{conv}} exp[K s^(1/2)] for large s and t less than a critical threshold t_c, where K is a t-dependent constant, and theta_{conv} is a critical exponent which does not change with t. We find theta_{conv} is 1/4 for the square lattice, but -1/4 for the hexagonal lattice. The reason for this unexpected non-universality of theta_{conv} is traced to existence of sharp corners in the asymptotic shape of these polygons.Comment: 8 pages, 5 figures, revtex

A Time-Dependent Classical Solution of C=1 String Field Theory and Non-Perturbative Effects

We describe a real-time classical solution of $c=1$ string field theory written in terms of the phase space density, $u(p,q,t)$, of the equivalent fermion theory. The solution corresponds to tunnelling of a single fermion above the filled fermi sea and leads to amplitudes that go as \exp(- C/ \gst). We discuss how one can use this technique to describe non-perturbative effects in the Marinari-Parisi model. We also discuss implications of this type of solution for the two-dimensional black hole.Comment: 23

Stringy Quantum Effects in 2-Dimensional Black-Hole

We discuss the classical 2-dim. black-hole in the framework of the non-perturbative formulation (in terms of non-relativistic fermions) of c=1 string field theory. We identify an off-shell operator whose classical equation of motion is that of tachyon in the classical graviton-dilaton black-hole background. The black-hole `singularity' is identified with the fermi surface in the phase space of a single fermion, and as such is a consequence of the semi-classical approximation. An exact treatment reveals that stringy quantum effects wash away the classical singularity.Comment: 17p, TIFR/TH/92-63; (v3) tex error correcte

Bethe approximation for a system of hard rigid rods: the random locally tree-like layered lattice

We study the Bethe approximation for a system of long rigid rods of fixed length k, with only excluded volume interaction. For large enough k, this system undergoes an isotropic-nematic phase transition as a function of density of the rods. The Bethe lattice, which is conventionally used to derive the self-consistent equations in the Bethe approximation, is not suitable for studying the hard-rods system, as it does not allow a dense packing of rods. We define a new lattice, called the random locally tree-like layered lattice, which allows a dense packing of rods, and for which the approximation is exact. We find that for a 4-coordinated lattice, k-mers with k>=4 undergo a continuous phase transition. For even coordination number q>=6, the transition exists only for k >= k_{min}(q), and is first order.Comment: 10 pages, 10 figure

Probing Type I' String Theory Using D0 and D4-Branes

We analyse the velocity-dependent potentials seen by D0 and D4-brane probes moving in Type I' background for head-on scattering off the fixed planes. We find that at short distances (compared to string length) the D0-brane probe has a nontrivial moduli space metric, in agreement with the prediction of Type I' matrix model; however, at large distances it is modified by massive open strings to a flat metric, which is consistent with the spacetime equations of motion of Type I' theory. We discuss the implication of this result for the matrix model proposal for M-theory. We also find that the nontrivial metric at short distances in the moduli space action of the D0-brane probe is reflected in the coefficient of the higher dimensional v^4 term in the D4-brane probe action.Comment: 12 pages, latex. References added and some typos correcte

Non-relativistic Fermions, Coadjoint Orbits of \winf\ and String Field Theory at c=1c=1

We apply the method of coadjoint orbits of \winf-algebra to the problem of non-relativistic fermions in one dimension. This leads to a geometric formulation of the quantum theory in terms of the quantum phase space distribution of the fermi fluid. The action has an infinite series expansion in the string coupling, which to leading order reduces to the previously discussed geometric action for the classical fermi fluid based on the group $w_\infty$ of area-preserving diffeomorphisms. We briefly discuss the strong coupling limit of the string theory which, unlike the weak coupling regime, does not seem to admit of a two dimensional space-time picture. Our methods are equally applicable to interacting fermions in one dimension.Comment: 22 page

Classical Fermi Fluid and Geometric Action for c=1c=1

We formulate the $c=1$ matrix model as a quantum fluid and discuss its classical limit in detail, emphasizing the $\hbar$ corrections. We view the fermi fluid profiles as elements of \winf-coadjoint orbit and write down a geometric action for the classical phase space. In the specific representation of fluid profiles as `strings' the action is written in a four-dimensional form in terms of gauge fields built out of the embedding of the `string' in the phase plane. We show that the collective field action can be derived from the above action provided one restricts to quadratic fluid profiles and ignores the dynamics of their `turning points'.Comment: 31 pages. (Revised version