Entropic exponents of grafted lattices stars

Abstract

The surface entropic exponents of half-space lattice stars grafted at their central nodes in a hard wall are estimated numerically using the PERM algorithm. In the square half-lattice the exact values of the exponents are verified, including Barber's scaling relation and a generalisation for $2$-stars with one and two surface loops respectively. This is the relation \[ γ_{211}=2\,γ_{21}-γ_{20},\] where ${\gamma }_{21}$ and ${\gamma }_{211}$ are the surface entropic exponents of a grafted $2$-star with one and two surface loops respectively, and ${\gamma }_{20}$ is the surface entropic exponent with no surface loops. This relation is also tested in the cubic half-lattice where surface entropic exponents are estimated up to $5$-stars, including many with one or more surface loops. Barber's scaling relation and the relation \[ γ_{3111}=γ_{30}-3\,γ_{31}+3\,γ_{311} \] are also tested, where the exponents $\{{\gamma }_{31},{\gamma }_{311},{\gamma }_{3111}\}$ are of grafted $3$-stars with one, two or three surface loops respectively, and ${\gamma }_{30}$ is the surface exponent of grafted $3$-stars.

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