233 research outputs found
Character Expansion Methods for Matrix Models of Dually Weighted Graphs
We consider generalized one-matrix models in which external fields allow
control over the coordination numbers on both the original and dual lattices.
We rederive in a simple fashion a character expansion formula for these models
originally due to Itzykson and Di Francesco, and then demonstrate how to take
the large N limit of this expansion. The relationship to the usual matrix model
resolvent is elucidated. Our methods give as a by-product an extremely simple
derivation of the Migdal integral equation describing the large limit of
the Itzykson-Zuber formula. We illustrate and check our methods by analyzing a
number of models solvable by traditional means. We then proceed to solve a new
model: a sum over planar graphs possessing even coordination numbers on both
the original and the dual lattice. We conclude by formulating equations for the
case of arbitrary sets of even, self-dual coupling constants. This opens the
way for studying the deep problem of phase transitions from random to flat
lattices.Comment: 22 pages, harvmac.tex, pictex.tex. All diagrams written directly into
the text in Pictex commands. (Two minor math typos corrected.
Acknowledgements added.
Induced QCD at Large N
We propose and study at large N a new lattice gauge model , in which the
Yang-Mills interaction is induced by the heavy scalar field in adjoint
representation. At any dimension of space and any the gauge fields can be
integrated out yielding an effective field theory for the gauge invariant
scalar field, corresponding to eigenvalues of the initial matrix field. This
field develops the vacuum average, the fluctuations of which describe the
elementary excitations of our gauge theory. At we find two phases
of the model, with asymptotic freedom corresponding to the strong coupling
phase (if there are no phase transitions at some critical ). We could not
solve the model in this phase, but in the weak coupling phase we have derived
exact nonlinear integral equations for the vacuum average and for the scalar
excitation spectrum. Presumably the strong coupling equations can be derived by
the same method.Comment: 20 page
Boundary changing operators in the O(n) matrix model
We continue the study of boundary operators in the dense O(n) model on the
random lattice. The conformal dimension of boundary operators inserted between
two JS boundaries of different weight is derived from the matrix model
description. Our results are in agreement with the regular lattice findings. A
connection is made between the loop equations in the continuum limit and the
shift relations of boundary Liouville 3-points functions obtained from Boundary
Ground Ring approach.Comment: 31 pages, 4 figures, Introduction and Conclusion improve
Loop Gas Model for Open Strings
The open string with one-dimensional target space is formulated in terms of
an SOS, or loop gas, model on a random surface. We solve an integral equation
for the loop amplitude with Dirichlet and Neumann boundary conditions imposed
on different pieces of its boundary. The result is used to calculate the mean
values of order and disorder operators, to construct the string propagator and
find its spectrum of excitations. The latter is not sensible neither to the
string tension \L nor to the mass of the ``quarks'' at the ends of the
string. As in the case of closed strings, the SOS formulation allows to
construct a Feynman diagram technique for the string interaction amplitudes
Rational Theories of 2D Gravity from the Two-Matrix Model
The correspondence claimed by M. Douglas, between the multicritical regimes
of the two-matrix model and 2D gravity coupled to (p,q) rational matter field,
is worked out explicitly. We found the minimal (p,q) multicritical potentials
U(X) and V(Y) which are polynomials of degree p and q, correspondingly. The
loop averages W(X) and \tilde W(Y) are shown to satisfy the Heisenberg
relations {W,X} =1 and {\tilde W,Y}=1 and essentially coincide with the
canonical momenta P and Q. The operators X and Y create the two kinds of
boundaries in the (p,q) model related by the duality (p,q) - (q,p). Finally, we
present a closed expression for the two two-loop correlators and interpret its
scaling limit.Comment: 24 pages, preprint CERN-TH.6834/9
Anomalous dimension and local charges
AdS space is the universal covering of a hyperboloid. We consider the action
of the deck transformations on a classical string worldsheet in . We argue that these transformations are generated by an infinite linear
combination of the local conserved charges. We conjecture that a similar
relation holds for the corresponding operators on the field theory side. This
would be a generalization of the recent field theory results showing that the
one loop anomalous dimension is proportional to the Casimir operator in the
representation of the Yangian algebra.Comment: 10 pages, LaTeX; v2: added explanations, reference
Thermodynamics of 2D string theory
We calculate the free energy, energy and entropy in the matrix quantum
mechanical formulation of 2D string theory in a background strongly perturbed
by tachyons with the imaginary Minkowskian momentum
(``Sine-Liouville'' theory). The system shows a thermodynamical behaviour
corresponding to the temperature . We show that the
microscopically calculated energy of the system satisfies the usual
thermodynamical relations and leads to a non-zero entropy.Comment: 13 pages, lanlmac; typos correcte
Plane wave limit of local conserved charges
We study the plane wave limit of the Backlund transformations for the
classical string in AdS space times a sphere and obtain an explicit expression
for the local conserved charges. We show that the Pohlmeyer charges become in
the plane wave limit the local integrals of motion of the free massive field.
This fixes the coefficients in the expansion of the anomalous dimension as the
sum of the Pohlmeyer charges.Comment: v2: added explanation
On the commuting charges for the highest dimension SU(2) operators in planar SYM
We consider the highest anomalous dimension operator in the SU(2) sector of
planar SYM at all-loop, though neglecting wrapping contributions.
In any case, the latter enter the loop expansion only after a precise
length-depending order. In the thermodynamic limit we write both a linear
integral equation for the Bethe root density and a linear system obeyed by the
commuting charges. Consequently, we determine the leading strong coupling
contribution to the density and from this an approximation to the leading and
sub-leading terms of any charge : it scales as , which
generalises the Gubser-Klebanov-Polyakov energy law. In the end, we briefly
extend these considerations to finite lengths and 'excited' operators by using
the idea of a non-linear integral equation.Comment: Latex file, 20 pages, some typos corrected, some technical details
expanded and explaine
Yang-Lee Zeros of the Two- and Three-State Potts Model Defined on Feynman Diagrams
We present both analytic and numerical results on the position of the
partition function zeros on the complex magnetic field plane of the
(Ising) and states Potts model defined on Feynman diagrams
(thin random graphs). Our analytic results are based on the ideas of
destructive interference of coexisting phases and low temperature expansions.
For the case of the Ising model an argument based on a symmetry of the saddle
point equations leads us to a nonperturbative proof that the Yang-Lee zeros are
located on the unit circle, although no circle theorem is known in this case of
random graphs. For the states Potts model our perturbative results
indicate that the Yang-Lee zeros lie outside the unit circle. Both analytic
results are confirmed by finite lattice numerical calculations.Comment: 16 pages, 2 figures. Third version: the title was slightly changed.
To be published in Physical Review
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