DOUBLE SCALING LIMIT IN O(N) VECTOR MODELS IN D DIMENSIONS

1992 ◽  
Vol 07 (07) ◽  
pp. 1391-1413 ◽  
Author(s):  
P. DI VECCHIA ◽  
M. KATO ◽  
N. OHTA
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Scaling Limit ◽  
Coupling Constant ◽  
Polymer Model ◽  
Critical Theories ◽  
A Value ◽  
Arbitrary Potential ◽  
N Vector ◽  
Double Scaling Limit

Using the standard 1/N expansion, we study O(N) vector models in D dimensions with an arbitrary potential. We limit ourselves to renormalizable theories. We show that there exists a value of the coupling constant corresponding to a critical point and that a double scaling limit can be performed as in D=0 and in the case of matrix models in D=0, 1. For D=1 the theory is renormalizable with an arbitrary potential and we find in general a hierarchy of critical theories labeled by an integer k. The universal partition function obtained in the double scaling limit is constructed. Finally, we show that the critical behaviour of those models is the same as a branched polymer model recently constructed by Ambjørn, Durhuus and Jónsson.

scholarly journals FROM ONE-MATRIX MODEL TO KONTSEVICH MODEL

1993 ◽  
Vol 08 (30) ◽  
pp. 2875-2890 ◽  
Author(s):  
J. AMBJØRN ◽  
C. F. KRISTJANSEN
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Matrix Model ◽  
Correlation Functions ◽  
Hermitian Matrix ◽  
Scaling Limit ◽  
Genus Zero ◽  
Kdv Hierarchy ◽  
Hermitian Matrix Model ◽  
Double Scaling Limit

Loop equations of matrix models express the invariance of the models under field redefinitions. We use loop equations to prove that it is possible to define continuum times for the generic Hermitian one-matrix model such that all correlation functions in the double scaling limit agree with the corresponding correlation functions of the Kontsevich model expressed in terms of KdV times. In addition the double scaling limit of the partition function of the Hermitian matrix model agrees with the τ-function of the KdV hierarchy corresponding to the Kontsevich model (and not the square of the τ-function) except for some complications at genus zero.

scholarly journals THE CRITICAL POINT OF UNORIENTED RANDOM SURFACES WITH A NONEVEN POTENTIAL

1993 ◽  
Vol 08 (06) ◽  
pp. 1139-1152
Author(s):  
M.A. MARTÍN-DELGADO
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Critical Point ◽  
Discrete Model ◽  
Recursion Relation ◽  
Scaling Limit ◽  
Random Surfaces ◽  
Matrix Ensemble ◽  
Quartic Interaction ◽  
Double Scaling Limit

The discrete model of the real symmetric one-matrix ensemble is analyzed with a cubic interaction. The partition function is found to satisfy a recursion relation that solves the model. The double scaling-limit of the recursion relation leads to a Miura transformation relating the contributions to the free energy coming from oriented and unoriented random surfaces. This transformation is the same kind as found with a quartic interaction.

scholarly journals NON-PERTURBATIVE SOLUTION OF THE SUPER-VIRASORO CONSTRAINTS

1993 ◽  
Vol 08 (13) ◽  
pp. 1205-1214 ◽  
Author(s):  
K. BECKER ◽  
M. BECKER
Keyword(s):  
Partition Function ◽  
Matrix Model ◽  
Scaling Limit ◽  
Virasoro Constraints ◽  
Perturbative Solution ◽  
The One ◽  
Genus Expansion ◽  
Double Scaling Limit

We present the solution of the discrete super-Virasoro constraints to all orders of the genus expansion. Integrating over the fermionic variables we get a representation of the partition function in terms of the one-matrix model. We also obtain the non-perturbative solution of the super-Virasoro constraints in the double scaling limit but do not find agreement between our flows and the known supersymmetric extensions of KdV.

scholarly journals ON THE CONTINUUM LIMIT OF THE CONFORMAL MATRIX MODELS

1993 ◽  
Vol 08 (18) ◽  
pp. 3107-3137 ◽  
Author(s):  
A. MIRONOV ◽  
S. PAKULIAK
Keyword(s):  
Matrix Models ◽  
Continuum Limit ◽  
Scaling Limit ◽  
Universality Class ◽  
Partition Functions ◽  
Discrete Level ◽  
New Class ◽  
Double Scaling Limit ◽  
The Continuum

The double scaling limit of a new class of the multi-matrix models proposed in Ref. 1, which possess the W-symmetry at the discrete level, is investigated in detail. These models are demonstrated to fall into the same universality class as the standard multi-matrix models. In particular, the transformation of the W-algebra at the discrete level into the continuum one of the papers2 is proposed and the corresponding partition functions compared. All calculations are demonstrated in full in the first nontrivial case of W(3)-constraints.

scholarly journals SUPERLOOP EQUATIONS AND TWO-DIMENSIONAL SUPERGRAVITY

1992 ◽  
Vol 07 (21) ◽  
pp. 5337-5367 ◽  
Author(s):  
L. ALVAREZ-GAUMÉ ◽  
H. ITOYAMA ◽  
J.L. MAÑES ◽  
A. ZADRA
Keyword(s):  
Partition Function ◽  
Discrete Model ◽  
Continuum Limit ◽  
Basic Assumption ◽  
Scaling Limit ◽  
Integrable Hierarchy ◽  
Two Dimensional ◽  
Virasoro Constraints ◽  
Double Scaling Limit ◽  
The Continuum

We propose a discrete model whose continuum limit reproduces the string susceptibility and the scaling dimensions of (2, 4m) minimal superconformal models coupled to 2D supergravity. The basic assumption in our presentation is a set of super-Virasoro constraints imposed on the partition function. We recover the Neveu-Schwarz and Ramond sectors of the theory, and we are also able to evaluate all planar loop correlation functions in the continuum limit. We find evidence to identify the integrable hierarchy of nonlinear equations describing the double scaling limit as a supersymmetric generalization of KP studied by Rabin.

scholarly journals AN OPERATOR FORMALISM FOR UNITARY MATRIX MODELS

1991 ◽  
Vol 06 (29) ◽  
pp. 2727-2739 ◽  
Author(s):  
K. N. ANAGNOSTOPOULOS ◽  
M. J. BOWICK ◽  
N. ISHIBASHI
Keyword(s):  
Orthogonal Polynomials ◽  
Matrix Models ◽  
Pseudodifferential Operators ◽  
Scaling Limit ◽  
Unitary Matrix ◽  
String Equation ◽  
Operator Formalism ◽  
Multicritical Point ◽  
Compact Formulation ◽  
Double Scaling Limit

We analyze the double scaling limit of unitary matrix models in terms of trigonometric orthogonal polynomials on the circle. In particular we find a compact formulation of the string equation at the kth multicritical point in terms of pseudodifferential operators and a corresponding action principle. We also relate this approach to the mKdV hierarchy which appears in the analysis in terms of conventional orthogonal polynomials on the circle.

Branched polymers from a double-scaling limit of matrix models

1991 ◽  
Vol 360 (2-3) ◽  
pp. 463-479 ◽  
Author(s):  
Arlen Anderson ◽  
Robert C. Myers ◽  
Vipul Periwal
Keyword(s):  
Matrix Models ◽  
Scaling Limit ◽  
Branched Polymers ◽  
Double Scaling Limit

NEW CRITICAL BEHAVIORS FOR ONE-HERMITIAN-MATRIX MODELS

1992 ◽  
Vol 07 (07) ◽  
pp. 1527-1551
Author(s):  
P.M.S. PETROPOULOS
Keyword(s):  
Matrix Models ◽  
Hermitian Matrix ◽  
Scaling Limit ◽  
Field Theories ◽  
Polynomial Coefficients ◽  
Multicritical Points ◽  
Exact Correlation Functions ◽  
Genus One ◽  
Exact Correlation ◽  
Double Scaling Limit

In the general framework of one-Hermitian-matrix models, we study critical behaviors such that δx~δRm/n~δSm; δx, δS and δR are, respectively, the bare cosmological constant and the orthogonal-polynomial coefficients around criticality. On the sphere, we prove the existence of consistent multicriticality conditions such that string equations exhibit the above behavior. We define a double scaling limit and write down exact equations for the specific heat for any (m, n) model. Their solutions are unambiguous and the only corrections come from genus-one topology. We compute exact correlation functions for well-defined scaling operators. These belong to two different sectors. One of them is such that any squared operator vanishes when inserted in any correlation function. We discuss briefly the flows between these multicritical points as well as the nature of the 2D field theories coupled to gravity which they can describe.

N = ½ SUPERSTRING IN ZERO DIMENSION AND THE SPONTANEOUS BREAKDOWN OF THE SUPERSYMMETRY

1992 ◽  
Vol 07 (32) ◽  
pp. 2979-2989 ◽  
Author(s):  
SHIN’ICHI NOJIRI
Keyword(s):  
Matrix Models ◽  
Random Matrix ◽  
Continuum Limit ◽  
Scaling Limit ◽  
Partition Functions ◽  
Spontaneous Breakdown ◽  
Random Matrix Models ◽  
Double Scaling Limit ◽  
The Continuum ◽  
String Theories

We propose random matrix models which have N=1/2 supersymmetry in zero dimension. The supersymmetry breaks down spontaneously. It is shown that the double scaling limit can be defined in these models and the breakdown of the supersymmetry remains in the continuum limit. The exact non-trivial partition functions of the string theories corresponding to these matrix models are also obtained.

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