Entropic exponents of grafted lattice 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 generalization for 2-stars with one and two surface loops respectively. This is the relation γ_{211}=2γ_{21}-γ_{20}, where γ_{21} and γ_{211} are the surface entropic exponents of a grafted 2-star with one and two surface loops, respectively, and γ_{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}-3γ_{31}+3γ_{311} are also tested, where the exponents {γ_{31},γ_{311},γ_{3111}} are of grafted 3-stars with one, two, or three surface loops, respectively, and γ_{30} is the surface exponent of grafted 3-stars.