NONUNIVERSAL PROPERTIES OF SELF-INTERACTING POLYMER IN NON-HOMOGENEOUS ENVIRONMENT MODELED BY 3-SIMPLEX FRACTAL LATTICE

We have studied lattice self-avoiding polygons with attractive interaction between contacts which are nonconsecutively visited nearest neighboring sites. The lattice of choice is 3-simplex fractal lattice and the model represents a ring polymer in non-homogeneous solution whose quality depends on the interaction parameter. It has already been shown, by the renormalization group approach, that polymer on this lattice at any nonzero temperature can exist only in the extended phase. Universal critical exponents, which do not depend on the interaction strength, have also been determined. In this paper we are concerned with two nonuniversal quantities: the connectivity constant related with the free energy of the model and the mean number of contacts related with the internal energy. We have shown that the connectivity constant is an unboundedly increasing function of the interaction strength, while the mean number of contacts is an increasing function asymptotically approaching a limiting value equal to one half, which is the mean number of contacts in the case of Hamiltonian walks on the same lattice. This limiting value is expected, since in the limit of infinite interaction (or zero temperature) each self-avoiding walk on 3-simplex lattice becomes maximally compact and occupies all lattice points, i.e. becomes Hamiltonian walk.

First-passage percolation on the square lattice. III

The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant, Lp and (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.

First-passage percolation on the square lattice. III

The principal results of this paper concern the asymptotic behavior of the number of arcs in the optimal routes of first-passage percolation processes on the square lattice. Assuming that the underlying distribution has an atom at zero less than λ–1, where λ is the connectivity constant, Lp and (in some cases) almost sure convergence theorems are proved for the normalized route length processes. The proofs involve the extension of much of the existing theory of first-passage percolation to the case where negative time coordinates are permitted.