On the universality of matrix models for random surfaces

We present an alternative procedure to eliminate irregular contributions in the perturbation expansion of –matrix models representing the sum over triangulations of random surfaces, thereby reproducing the results of Tutte [1] and Brézin et al. [2] for the planar model. The advantage of this method is that the universality of the critical exponents can be proven from general features of the model alone without explicit determination of the free energy and therefore allows for several straightforward generalizations including cases with non-vanishing central charge . Received: 9 Novemver 1998 / Published online: 1 March 1999     

The random lattice as a regularization scheme

A semi-analytic method to compute the first coefficients of the renormalization group functions on a random lattice is introduced. It is used to show that the two-dimensional O(N) nonlinear σ-model regularized on a random lattice has the correct continuum limit. A degree κ of “randomness” in the lattice is introduced and an estimate of the ratio Λ_random/Λ_regular for two rather opposite values of κ in the σ-model is also given. This ratio turns out to depend on κ.     

Triangulated random surfaces

A model of discretized random surfaces that contains the extrinsic curvature as well as the usual area term in the action is considered. The renormalization group predicts that at large distances the model is indistinguishable from previous proposals of triangulated surfaces that contained only the area term, but, unlike them, does not grow spikes. The partition function and all its moments are finite and well defined. The model is solved for large d in the vicinity of the IR fixed point. The Hausdorff dimension is ∞ and the entropy exponent agrees with the one obtained by Zamolodchikov and others for the Polyakov action in the continuum.     

Self-affine random surfaces

We consider general d-dimensional random surfaces that are characterized by power-law power spectra defined in both infinite and finite spectral regions. The first type of surfaces belongs to the class of ideal fractals, whereas the second possess both the smallest and the largest scales and physically is more realistic. For both types we calculate the structure functions (SF) exactly; in addition for the second type we obtain the SF's asymptotic expansions. On this basis we show that the surfaces are (in statistical sense) self-affine and approximately self-affine, respectively. Depending on the value of the spectral exponent, we find imbalance between the finite size effects which results in systematic discrepancy in the scaling properties between the two types of surfaces. Explicit expressions for the topothesy, and in the case of second type of surfaces for the large correlation length and cross-over distances are also derived. Received 3 October 2001 / Received in final form 5 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: oyordanov@aubg.bg     

Retention capacity of random surfaces

We introduce a "water retention" model for liquids captured on a random surface with open boundaries and investigate the model for both continuous and discrete surface heights 0,1,…,n-1 on a square lattice with a square boundary. The model is found to have several intriguing features, including a nonmonotonic dependence of the retention on the number of levels: for many n, the retention is counterintuitively greater than that of an (n+1)-level system. The behavior is explained using percolation theory, by mapping it to a 2-level system with variable probability. Results in one dimension are also found.     

Bilocal Dynamics for Self-Avoiding Walks

We introduce several bilocal algorithms for lattice self-avoiding walks. We discuss their ergodicity in different confined geometries, for instance in strips and in slabs. A short discussion of the dynamical properties in the absence of interactions is given.     

Nonlocal structures: Bilocal photon

As a starting point, it is postulated that all particles and fields are built from a single primitive field, which must then be a massless fermion with a spin of one-half. Two helicities are embodied in a spin of one-half. The vacuum is an open Fermi sea whose height is a wave number . Elementary particles are structures having the form of standing-wave systems floating on the vacuum sea, with the height providing both the scale of inner structural size and the mass unit for the elementary particle mass spectrum. A bilocal photon starts with a function describing two primitive quanta with parallel spin and opposite spin. A centroid-time wave equation then couples-in an infinite set of orthogonal functions. The introduction of an operatorQ permits the reduction of the infinite secular determinant to a finite six-by-six determinant. Solutions (for the infinite expansion) are obtained describing photons with right-handed and left-handed polarizations. Superpositions of these give linearly polarized photons. Electric and magnetic field vectors, satisfying the vacuum Maxwell equations, are obtained from a bilocal Hertz vector given by = (2/³ c)(/t _r)_r(1,2), where (1,2) is the bilocal wave function, and t_r and r are the relative time and relative position variables.     

Bilocal lie groups and solitons

A group theoretical scheme where solition equations are associated with the integrability conditions for differential system is proposed. These conditions ensure the existence of a bilocal Lie group structure which naturally generates a set of conserved currents for arbitrary space-time dimensions.     

Multicritical points of unoriented random surfaces

Unoriented surfaces generated by real symmetric one-matrix models are solved in the scaling limit in which the size of the matrix (related to the string coupling constant) goes to infinity and the cosmological constant approaches a multicritical point of a suitably chosen potential. The solution involves skew orthogonal polynomials, and in spite of the non-local character of the operations d/dx or multiplication by x acting on these polynomials, a local differential formalism is shown to be present in this problem as well. The Gel'fand-Dikii pseudo-differential operator appears here factorized as a product of two differential operators of degrees m and (m − 1) respectively. The relations with other ensembles of random matrices are examined and the difficulties associated with multi-matrix models are pointed out.     

The Casimir pressure regularization in two-dimensional field models

A method for Casimir pressure calculation with the help of the regular part of the Green surface function is considered in the two-dimensional case. Also, a method for the approximate calculation of the regular part of the Green surface function using a Born-type series is suggested. It is tested for a problem for which the exact solution is known.     

Two regularization types as two different Nambu-Iona-Lasinio models

The Nambu-Iona-Lasinio models with 4-dimensional cutting and dimensional-analytical regularization types are compared. It is demonstrated that they describe two different models of light quark interaction. In the average-field approximation, the behavior of the scalar amplitude differs in the threshold region. Unlike the 4-dimensional cutting regularization in which the pole term corresponding to a sigma-meson can be separated near the threshold, the singularity of the scalar amplitude in the dimensional-analytical regularization is non-pole; moreover, it disappears completely for a certain value of the regularization parameter. One more significant difference between the two models is in the first-order expansion of the average field. The calculated meson contributions to the quark chiral condensate and dynamic quark mass demonstrate that despite their relative smallness, they can destabilize the Nambu-Iona-Lasinio model with 4-dimensional cutting regularization. On the contrary, the model with dimensional-analytical regularization is stabilized, which is manifested through a shift of regularization parameter values toward the stability region in which the contributions themselves decrease. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 20–31, April, 2006.