This paper is concerned with recessive and dominant solutions for the nonoscillatory second-order half-linear difference equations \[ \Delta(a_{n}\Phi(x_{n}))+b_{n}\Phi(x_{n+1})=0, \] where \(\Delta x_{n}=x_{n+1}-x_{n}\), \(\Phi(u)=| u| ^{p-2}u\) with \(p>1\), and \(\{a_{n}\},\{b_{n}\}\) are positive real sequences for \(n\geq1\). By using a uniqueness result for the zero-convergent solutions satisfying a suitable final condition, the authors prove that recessive solutions are the ``smallest solutions in a neighborhood of infinity'', as in the linear case. Other asymptotic properties of recessive and dominant solutions are also treated.