Milan Knežević

Adsorption of a flexible self-avoiding polymer chain : exact results on fractal lattices

The large-scale behavior of surface-interacting self-avoiding polymer chains placed on finitely ramified fractal lattices is studied using exact recursion relations. It is shown how to obtain surface susceptibility critical indices and how to modify a scaling relation for these indices in the case of fractal lattices. We present the exact results for critical exponents at the point of adsorption transition for polymer chains situated on a class of Sierpinski gasket-type fractals. We provide numerical evidence for a critical behavior of the type found recently in the case of bulk self-avoiding random walks at the fractal to Euclidean crossover.

Lattice animals on a class of hierarchical graphs

We study the statistics of lattice animals on a class of hierarchical graphs whose members can be labeled by a set of integers , g≥1. We have shown that the animal critical behavior crucially depends on the minimal value of these parameters. For we find the usual power-law behavior, while for the associated generating function displays an essential singularity ~exp[c(tc−t)−ψ], where c is a constant and the exponent ψ is related to the leading correction term in the asymptotic behavior of the number of animal configurations having N bonds, , ω = ψ/(ψ+1). We express the entropic exponent ω and the animal size exponent in terms of pertinent graph parameters.

Critical exponents of surface-interacting self-avoiding walks on a family of truncated n-simplex lattices

We study the critical behavior of surface-interacting self-avoiding random walks on a class of truncated simplex lattices, which can be labeled by an integer n ⩾ 3. Using the exact renormalization group method we have been able to obtain the exact values of various critical exponents for all values of n up to n = 6. We also derived simple formulas which describe the asymptotic behavior of these exponents in the limit of large n (n → ∞). In spite of the fact that the coordination number of the lattice tends to infinity in this limit, we found that most of the studied critical exponents approach certain finite values, which differ from corresponding values for simple random walks (without self-avoiding walk constraint).

Critical adsorption of random walks on fractal lattices with uniform coordination number

We study the problem of adsorption of random walks on a boundary of fractal lattices that have uniform coordination numbers. More specifically, for a suitable Gaussian model, situated on the Sierpinski gasket fractals with an interacting wall, we analyze critical properties using the renormalization group approach. In this way we have found exact expressions for a set of pertinent critical exponents. In particular, we have demonstrated that the crossover critical exponent, associated with the number of adsorbed monomers, can be expressed as simple combination of only three quantities—the end-to-end distance critical exponent, the substratum fractal dimension, and the adsorbing boundary fractal dimension.

Large scale behavior of a two-dimensional model of anisotropic branched polymers.

We study critical properties of anisotropic branched polymers modeled by semi-directed lattice animals on a triangular lattice. Using the exact transfer-matrix approach on strips of quite large widths and phenomenological renormalization group analysis, we obtained pretty good estimates of various critical exponents in the whole high-temperature region, including the point of collapse transition. Our numerical results suggest that this collapse transition belongs to the universality class of directed percolation.

Discrete spin cubic model on a fractal lattice - the ground state phase diagram

We study the ground state properties of the discrete spin cubic model situated on the Sierpinski gasket fractal lattice. The relevant model Hamiltonian contains exchange and quadrupolar interaction terms, associated with the parameters J1 and J2, respectively. We have found that the model under study has a very rich ground state diagram in the (J1,J2)-plane, with one ordered (ferromagnetic) state and three regions of different frustrated states.

A transfer-matrix study of directed lattice animals and directed percolation on a square lattice

We studied the large-scale properties of directed lattice animals and directed percolation on a square lattice. Using a transfer-matrix approach on strips of finite widths, we generated relatively long sequences of estimates for effective values of critical fugacity, percolation threshold and correlation length critical exponents. We applied two different extrapolation methods to obtain estimates for infinite systems. The precision of our final estimates is comparable to (or better than) the precision of the best currently available results.

The Yang–Lee edge singularity for the Ising model on two Sierpinski fractal lattices

We study the distribution of zeros of the partition function of the ferromagnetic Ising model near the Yang–Lee edge on two Sierpiski-type lattices. We have shown that relevant correlation length displays a logarithmic divergence near the edge, where Φ is a constant and δh distance from the edge, in the case of a modified Sierpinski gasket with a nonuniform coordination number. It is demonstrated that this critical behavior can be related to the critical behavior of a simple zero-field Gaussian model of the same structure. We have shown that there is no such connection between these two models on a second lattice that has a uniform coordination number. These findings suggest that fluctuations of the lattice coordination number of a nonhomogeneous self-similar structure exert the crucial influence on the critical behavior of both models.

Adsorption in a model of interacting branched polymers on a fractal lattice

We have studied a model of self-interacting branched polymers on the three-dimensional Sierpinski gasket lattice, in the presence of an attractive impenetrable fractal boundary. Using an exact renoramlization group approach, we have determined the phase diagram boundaries of this model, and its critical properties in different regions of the phase diagram.

Anisotropy in a fractal model of conductivity

Abstract Using an exact set of recursion relations we study the anisotropy of dc electrical conductivity on Dhars modified rectangular lattice of index p. It is found that in the limit of large values of p the asymptotic ratio of parallel to vertical macroscopic conductivity approaches the value 2p 2 3 . The opposite limit of low anisotropy, p ≈ 1, is studied by means of infinitesimal recursion relations. We discuss the influence of anisotropy on critical behavior of the model and make a comparison with some other lattice models.