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An adsorbing self-avoiding walk in the square lattice is illustrated in figure \[figure0\]. Let $\mathL^d$ denote the $d$-dimensional hypercubic lattice and denote a unit length edge with endpoints $\vec{x}$ and $\vec{y}$ in $\mathL^d$ by $\edge{\vec{x}}{\vec{y}}$. The hypercubic half-lattice $\mathL^d_+$ is defined by $$\mathL^d_+ = 
\{ \edge{\vec{x}}{\vec{y}} \in \mathL^d \vv 
\hbox{$\vec{x}(d) \geq 0$ and $\vec{y}(d) \geq 0$} \},$$ where $\vec{x}(d)$ is the $d$-th Cartesian component of $\vec{x}$, and $\vec{y}(d)$ is the $d$-th Cartesian component of $\vec{y}$. The boundary of $\mathL^d_+$ is given by $$\partial\mathL^d_+ = 
\{ \edge{\vec{x}}{\vec{y}} \in \mathL^d_+ \vv 
\hbox{$\vec{x}(d)=0$ and $\vec{y}(d)=0$} \},$$ and it is isomorphic to $\mathL^{d-1}$ if $d\geq 2$. Please note that these definitions, and the definitions of additional functions and quantities, are listed in table \[Defs\].

The number of self-avoiding walks of length $n$ from the origin in $\mathL^d$ is denoted by $c_n$. The *growth constant* of the self-avoiding walk [@H57; @H60] is defined by the limit $$\lim_{n\to\infty} c_n^{1/n} = \mu_d,$$ and $\kappa_d = \log \mu_d$ is the connective constant of the self-avoiding walk. This shows that $c_n = \mu_d^{n+o(n)}$. Self-avoiding walks from the origin in $\mathL^d_+$ are *positive walks*. The number of positive walks of length $n$ is denoted by $c_n^+$, and it is 