Generalized exponential function and discrete growth models

Here we show that a particular one-parameter generalization of the exponential function is suitable to unify most of the popular one-species discrete population dynamic models into a simple formula. A physical interpretation is given to this new introduced parameter in the context of the continuous Richards model, which remains valid for the discrete case. From the discretization of the continuous Richards’ model (generalization of the Gompertz and Verhulst models), one obtains a generalized logistic map and we briefly study its properties. Notice, however that the physical interpretation for the introduced parameter persists valid for the discrete case. Next, we generalize the (scramble competition) θ-Ricker discrete model and analytically calculate the fixed points as well as their stabilities. In contrast to previous generalizations, from the generalized θ-Ricker model one is able to retrieve either scramble or contest models.     

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 largeN 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 largeN limit of the Itzykson-Zuber formula. We illustrate and check our methods by analysing a number of models solvable by traditional means. We then proceed to solve a new model: a sum over planar graphys possessing even coordination numbers on both the original and the dual lattice. We conclude by formulating the equations for the case of arbitrary sets of even, self-dual coupling constants. This opens the way for studying the deep problems of phase transitions from random to flat lattices. January 1995     

Isosystolic inequalities and the topological expansion for random surface and matrix models

Using the isosystolic inequalities on Riemann surfaces, we prove that for many random surface or matrix models the radius of convergence of the perturbative series at fixed genus is independent of the genus. This result applies for instance to the dynamically triangulated random surface model in any dimension or to many matrix models with regular propagators in the superrenormalizable domain, for instance ³ in dimensiond<6, in dimensiond<4, and various otherP()₂ models (in particular all those containing an odd power of ). We hope that this result is a first step towards a more rigorous understanding of the genus dependence of surface models or of quantum gravity coupled with matter fields.     

Green's function and scattering matrix in a discrete oscillator basis

Convenient analytic finite-dimensional approximations for basic operators of scattering theory-specifically, the Green's function and the off-shell T matrix—are constructed in an oscillator basis for real-and complex-valued local and nonlocal interaction potentials. It is shown that the approximate operators converge smoothly to their exact counterparts as the dimensions of the oscillator basis are increased step by step. The simple and rather accurate formulas obtained in this study can be widely used in various applications of quantum scattering theory.     

A holomorphic and background independent partition function for matrix models and topological strings

We study various properties of a nonperturbative partition function which can be associated with any spectral curve. When the spectral curve arises from a matrix model, this nonperturbative partition function is given by a sum of matrix integrals over all possible filling fractions, and includes all the multi-instanton corrections to the perturbative 1/N$1/N$ expansion. We show that the nonperturbative partition function, which is manifestly holomorphic, is also modular and background independent: it transforms as the partition function of a twisted fermion on the spectral curve. Therefore, modularity is restored by nonperturbative corrections. We also show that this nonperturbative partition function obeys the Hirota equation and provides a natural nonperturbative completion for topological string theory on local Calabi–Yau 3-folds.     

Cluster expansion for abstract polymer models

A new direct proof of convergence of cluster expansions for polymer (contour) models is given in an abstract setting. It does not rely on Kirkwood-Salsburg type equations or combinatorics of trees. A distinctive feature is that, at all steps, the considered clusters contain every polymer at most once.     

Two hierarchies of integrable lattice equations associated with a discrete matrix spectral problem

Two hierarchies of nonlinear integrable positive and negative lattice models are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws of the positive hierarchy, then, the integrable coupling systems of the positive hierarchy are derived from enlarging Lax pair.     

On the supersymmetric partition function in QCD-inspired random matrix models

We show that the expression for the supersymmetric partition function of the chiral unitary (Laguerre) ensemble conjectured recently by Splittorff and Verbaarschot [Phys. Rev. Lett. 90, 041601 (2003)] follows from the general expression derived recently by Fyodorov and Strahov [J. Phys. A: Math. Gen. 36, 3203 (2003)].     

Optimized δ expansion for relativistic nuclear models

The optimized δ-expansion is a nonperturbative approach for field theoretic models which combines the techniques of perturbation theory and the variational principle. This technique is discussed in the λφ⁴ model and then implemented in the Walecka model for the equation of state of nuclear matter. The results obtained with the δ expansion are compared with those obtained with the traditional mean field, relativistic Hartree and Hartree-Fock approximations. Received: 17 March 1997 / Revised version: 27 August 1997     

Connected diagram expansion for lattice hamiltonian models

We present a connected diagram expansion technique for energy eigenvalues in Hamiltonian quantum theory. The technique should be particularly useful for lattice models of the type introduced by Wilson, Kogut and Susskind. The method is illustrated by the example of the quantum Ising model in 1 + 1 dimensions. A path integral formalism is outlined from which these results may be derived: it is based on the formal scattering theory of Gell-Mann and Goldberger.     

Advanced boundary conditions for eigenmode expansion models

In order to realise the full potential of eigenmode expansion models, advanced boundary conditions are required that can absorb the radiation impinging on the walls of the discretisation volume. In this paper, we will discuss and compare a number of these boundary conditions, like perfectly matched layers (PMLs), open (leaky mode) boundary conditions and transparent boundary conditions (TBCs). We will also introduce the case of PMLs with infinite absorption and discuss its relation to leaky mode expansion, leading to a deeper insight into the physics of PML.     

Advanced boundary conditions for eigenmode expansion models

In order to realise the full potential of eigenmode expansion models, advanced boundary conditions are required that can absorb the radiation impinging on the walls of the discretisation volume. In this paper, we will discuss and compare a number of these boundary conditions, like perfectly matched layers (PMLs), open (leaky mode) boundary conditions and transparent boundary conditions (TBCs). We will also introduce the case of PMLs with infinite absorption and discuss its relation to leaky mode expansion, leading to a deeper insight into the physics of PML.     

A systematic expansion for random Ising models

An approximation scheme for random Ising models on a square lattice is derived for the case of a symmetric probability distribution of the couplings. The calculation is explicitly performed for the binary distribution. It is suggested that the quenched free energy is an analytic function of the temperature down toT=0, and that a spin glass phase as a phase in the usual sense does not exist. Rigorous nontrivial bounds for the ground state energy and the ground state entropy are also obtained.Work performed within the research program of the Sonderforschungsbereich 125 Aachen-Jülich-Köln     

Low temperature expansion for continuous-spin Ising models

We consider general ferromagnetic spin systems with finite range interactions and an even single-spin distribution of compact support on IR. It is shown under mild assumptions on the single-spin distribution that a low temperature expansion, in powers ofT, for the free energy and the correlation functions is asymptotic. We also prove exponential clustering in the pure phases and analyticity of the free energy and of the correlation functions in the reciprocal temperature for Re large.Supported in part by N.S.F. Grant No. Phy. 78-15920 and MCS 78-01885On leave from: Départment de Physique Théorique, Université de Louvin, BelgiumSupported in part by the Swiss National Foundation for Scientific Research     

Physical solutions for unitary matrix models

We study the simplest double scaling limit of the integral over a unitary matrix, shown by Periwal and Shevitz to admit an exact solution in terms of the mKdV hierarchy. We show that there is a unique non-perturbative solution of the string equation corresponding to the true double scaling limit of the integral, which interpolates smoothly between weak and strong coupling regimes.     

The two-point correlation function of the one-dimensional matrix model

Following Kostov and Ben-Menahem, we calculate the two-puncture correlation function for the one-dimensional matrix model. We find that it depends on the details of discretization for all momenta p. Its only universal features are its vanishing as p → 0 and the appearance of double poles at |p| = n/√′, N = 1,2,…. We show how to derive these double poles in the conformal gauge treatment of Liouville gravity.