Entropic exponents of grafted lattices stars

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 $${\gamma }_{211}=2{\gamma }_{21}-{\gamma }_{20},$$ 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 $${\gamma }_{3111}={\gamma }_{30}-3{\gamma }_{31}+3{\gamma }_{311}$$ 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.