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

TL;DR

This paper analyzes the critical behavior of surface-interacting self-avoiding walks on truncated simplex lattices, deriving exact exponents for small n and asymptotic formulas for large n, revealing finite limits despite increasing lattice complexity.

Contribution

It provides exact critical exponents for self-avoiding walks on truncated simplex lattices and formulates their asymptotic behavior as n approaches infinity.

Findings

Critical exponents are obtained exactly for n up to 6.

Asymptotic formulas describe exponents for large n.

Most exponents approach finite limits as n increases.

Abstract

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 $n\ge 3$. 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 ($n\to\infty$). In spite of the fact that the coordination number of the lattice tends to infinity in this limit, we found that the most of the studied critical exponents approach certain finite values, which differ from corresponding values for simple random walks (without self-avoiding walk constraint).